Jorge António

In this thesis, we study different aspects of derived k-analytic geometry. Namely, we extend the theory of classical formal models for rigid k-analytic spaces to the derived setting. Having a theory of derived formal models at our disposal we proceed to study certain applications such as the representability of derived Hilbert stack in the derived k-analytic setting. We construct a moduli stack of derived k-adic representations of profinite spaces and prove its geometricity as a derived k-analytic stack. Under certain hypothesis we show the existence of a natural shifted symplectic structure on it. Our main applications is to study pro-étale k-adic local systems on smooth schemes in positive characteristic. Finally, we study at length an analytic analogue (both over the field of complex numbers C and over a non-archimedean field k) of the structured algebraic HKR, proved by Toen and Vezzosi.

in the 1-category dSt dAn C , ⌧ ét , P sm , where LocSys an C,n (X) and Conn an C,n (X) denote the analytification of both LocSys C,n (X) and Conn C,n (X), respectively.

First of all and above all, I would like to thank to my advisor, Bertrand Toën. Not only he guided me through this last three with immense devotion, but also shared many thoughts, ideas and knowledge, from which I much profited. His vision and deep knowledge of mathematics have had a great impact in each ways I pursue new ideas in mathematics. Not only, I must thank Bertrand for all the guidance, but also as a source of inspiration.

I also want to express my deep gratitude towards Marco Robalo and Mauro Porta. I met Marco for a quiet few years from now and since then he has been an important mathematical figure for me. He seems to always have reasonable advices and suggestions for the younger. From Mauro I learned a huge amount of mathematics. He grasps at length the most delicate technical details. Furthermore, his kindness sharing it with me helped me profoundly. This has been the case since I was a master student in Paris. Moreover, Mauro's enthusiasm for mathematics is enormous and contagious.

Carlos Simpson has been a great source of mathematical inspiration. His works on so many fields of mathematics and his vision of mathematics had a tremendous impact on the contents of the present thesis. I received great advice from Carlos when I visited him in Nice, a couple of years ago. Moreover, Carlos accepted to be a reporter of the current thesis. For all this, I am tremendously thankful.

Bhargav Bhatt has been a key figure in my mathematical background. Not only did I navigate through much of his work, which is always so interesting and full of content, but I also greatly profited from discussing personally with him. Bhargav always showed patiente and willingness to discuss mathematics with me. Having the possibility to engage in his way of thinking was a great opportunity to me as a mathematician. Moreover, Bhargav kindly accepted to write a report for the thesis. For all these reasons, I must express my deep gratitude to him.

I feel honored to have Carlos Simpson, Vadim Schechtman, Anna Cadoret and Tony Yu Yue as members of the jury of my thesis. Accordingly, I would like to thank them here.

I great benefited from the companion of many people in Toulouse. Firstly, I would like to Joseph Tapia, from which I learned so much. He has always been available to discuss mathematics and learn more about it. His vast knowledge in so many different disciplines of the subject had a great impact on me. He has been always humble towards me, as a student. Consequently I felt that my ideas and my development, as a mathematician, are appreciated. I would also like to thank to Joan Milles, Michel Vaquié, Marcello Bernadara and my PhD brothers Elena, Massimo and Leyth. It has been a great pleasure to discuss with Dario Beraldo and Tasos Moulinos. I am thankful not only for the mathematics but also from the cultural experience which this past three years have provided me.

I must also express my gratitude towards many mathematicians that I have met, around the globe, in the last three years. Not only these meetings provided me with immense knowledge, but also allowed me to be into contact with inspiring minds. I want thus to thank to Jacob Lurie, Gabriele Vezzosi, Mathieu Anel, Anthony Blanc, Samuel Bach Matthew Morrow, Peter Scholze, Benjamin Hennion, Adheel Kahn, Joost Nuiten, Ricardo Campos, Jean-Baptiste Teyssier, Pierre Schapira, François Petit, Ariane Mézard, Laurent Fargues, Dustin Clausen Vincent Daruvar, Emiliano Ambrosi, Valerio Melani, Brian Conrad, Benoit Stroh, Benoit Cadorel, Alberto Vezzani, David Gepner, Carl Wang-Erickson, Vincent Lafforgue, Marco d'Addezio, Marc Hoyois, Ben Antieu, Apart from the mathematical community, I had the luck to meet many great people. People whose way of living had an enormous influence in the developing of myself as an individual. I first want to thank to all my old friends from my hometown. Paulo Ilhéu, Joao Nuno, Joao Ferreira, Miguel Sabino, Margarida Santos, Marta Azenha, Lea Lopes, Pedro Duraes, Luís, Zézinho, Diogo Guedes. I also share great memories from the friends I met in Paris: Rodrigo Codorniu, Mattia Cavicchi, Nuno Marinho, Pedro Damas, Mariana Fabiao, Anita Novais, Fernando and Isabel, Mário Pereira, Bruno Vilela, Madalena Caetano, Sofia Balsinha, Rita Pereira, Eduardo. Thank you for such great moments I also had the great opportunity the opportunity to contribute from the great companion of all my friends in Toulouse. Guilherme Sousa, Beatriz Catarino, Marta Bruxelas, Pedro Velez, Rita Tojo.

Barbosa and Titá had always been a source of love and protection. They always supported me and accepted me as a son, since I was a child. I feel so grateful for all these years and so much more.

I could not feel other than deeply grateful to have met Catarina O'Neill, whose love and tenderness brought so much light and meaning to my life. You have always supported and appreciated my mathematical life. And above all, you always showed me the bright side of the moon. I am deeply thankful for such warm.

Lastly, I want to thank to my parents that always sacrifice so much for me and for my development. So many times without asking nothing in their favor. I am deeply touched by your devotion, as parents, towards me, as a 2 son.

Your influence made kept me moving forward. To you, obrigado por tudo! And as I close this section, I would like to say a few more words. The universe is vast and our understanding limited. Nonetheless, the inspiration we obtain by coming across extraordinary people and engaging in such vast intellectual activity is invaluable. I have been so lucky to make part of this journey.

Résumé

Dans cette thèse, on étude différents aspects de la théorie de la géométrie dérivée rigide analytique. D'abord, on étude et généralise le théorèome classique de localisation de Raynaud au cadre dérivé. Muni d'une théorie des modéles formels, développé dans cette thése, on étude ses applications à l'étude des certains espaces de modules dérivés. Certains exemples correspondent bien au champ d'Hilbert rigide analytique dérivé et le champ des représentations continues des groupes fondamentales des variétés lisses sur un corps fini. La structure dérivée sur ce dernier nos permet de comprendre totalement la théorie de déformations des représentations galoisiennes. Enfin, on montre que ce dernier admet une structure sympléctique dérivé naturel. Ce dernier résultat s'appuye dans le théorème de HKR en géométrie analytique qui on prouve en collaboration avec F. Petit et M. Porta.

Liste des résultats principaux

Soit X un schéma propre et lisse sur un corps algébriquement clos. On est intéressé à l'étude des systémes locaux `-adiques étales sur X, d'un certain rang. En effet, d'après les travaux de V. Drinfeld et plus récemment de V. Lafforgue autour de la correspondance de Langlands pour les corps de fonctions on sait que la famille de tels objets est fortément réliée aux formes automorphes. Il semble donc naturel d'étudier le foncteur de modules qui paramètre des systémes locaux `-adiques sur X ou, de manière équivalente, des représentations continues `-adiques du groupe fondamentale étale, ⇡ ét 1 (X). Tel foncteur est noté LocSys `,n (X) : Afd op Q `! Grpd, où Afd op Q `denote la catégorie des Q `-algèbres affinoides et Grpd la catégorie des groupoides, et il associe à chaque Q `-algébre affinoide A 2 Afd op Q `7 ! LocSys `,n (X)(A) 2 Grpd, où LocSys `,n (X)(A) correspondant au groupoide des systèmes locaux `-adiues étales sur X. Le principale but de cette thèse est l'étude des propriétés géométriques de LocSys `,n (X). En particulier, on prouve le thérèome: (X) paramètre des systèmes locaux, sur X, munis d'une trivialisation. En plus, LocSys `,n (X) admet une structure dérivée naturelle tel que si ⇢ 2 LocSys `,n (X)(Q `) alors le complexe cotangent analytique est donné par L an LocSys `,n (X),⇢ ' C ⇤ ét (X, Ad⇢) _ [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF],

où Ad(⇢) := ⇢ ⌦ ⇢ _ c'est la répresentation adjointe de ⇢.

Un autre résultat important concernant la géométrie de LocSys `,n (X) est l'existence d'une forme sympléctique décalée sur LocSys `,n (X): Theorem 2. Soit X un schéma propre et lisse sur un corps algébriquement clos de dimension d. Le champ Q `- analytique LocSys `,n (X) admet une forme sympléctique (2d 2)-décalée naturelle !. En plus, la (2d 2)-forme sur LocSys `,n (X) est induite de la dualité de Poincaré en cohomologie étale.

Pour prouver l'existence d'une telle structure dérivée naturelle sur LocSys `,n (X) on a eu besoin de généraliser le théorème classique de localisation de Raynaud au cadre dérivé. Plus précisement, on a prouvé Theorem 3. Soit k un corps non-archimédien dont la valuation est de rang 1 et k son anneau des entiers. Soit dAn k la 1-catégorie des espaces k-analytiques dérivés et dfSch l'1-catégorie des schémas formels dérivés, de type (topologiquement) fini, sur k . Alors il existe un foncteur de rigidification ( ) rig : dfSch k ! dAn k dont la restriction aux schémas formels discrets coincide avec le foncteur de rigidification de Raynaud usuel. En plus, on a une équivalence d'1-catégories

( ) rig : dfSch k [S 1
] ! dAn k où S denote la classe des éclatements dérivés admissibles et génériquement strong dans dfSch k .

Le thérèome de Raynaud dérivé a trouvé jusqu'à maintenant certaines applications importantes. Un exemple c'est le prochain résultat prouvé en collaboration avec Mauro Porta: Theorem 4. Soit X 2 dAn k et X 2 dfSch k tel que (X) rig ' X dans dAn k . Alors on a une suite exacte d'1-catégories stables Coh + nil (X) ! Coh + (X) ! Coh + (X), où Coh + nil (X) denote la sous-catégorie pleine de Coh + (X) engendré par les modules presque-parfaits sur X qui sont supportés dans la fibre spéciale, X sp , de X.

Enfin, on a prouvé un analogue rigide k-analytique et aussi C-analytique du théorème de HKR structuré, qui était prouvé par B. Toën et G. Vezzosi. Ce théorème fait partir d'un travail en collaboration avec F. Petit et M. Porta:

Theorem 5. Soit k le corps des nombres complexes, C, ou un corps non-archimédien de caractéristique 0 de valuation non-triviale. Soit X un espace k-analytique dérivé. Alors il existe une équivalence des espaces kanalytiques dérivés X X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF], compatibles avec la projection vers X.

Introduction

Let X be a smooth and proper scheme over an algebraically closed field. One usually is interested in studying `-adic étale local systems on X. For example, after the works of V. Drinfeld and more recently V. Lafforgue on geometric Langlands correspondence for function fields one can relate such arithmetic objects to automorphic forms on X. Therefore, it seems natural to study the moduli functor parametrizing étale `-adic local systems on X. Such moduli can be described as a functor LocSys `,n (X) : Afd op k ! Grpd given on objects by the formula

A 2 Afd op k 7 ! LocSys `,n (X)(A) 2 Grpd
where LocSys `,n (X)(A) 2 Grpd denotes the groupoid of étale A-adic étale local systems on X, Afd op k denotes the category of affinoid k-algebras (k a finite extension of Q `) and Grpd denotes the category of groupoids. A main motivation of the current thesis was the study of the geometric properties of LocSys `,n (X). In particular, we prove the following theorem: Theorem 1.2.1.1. Let X be a smooth and proper scheme. Then the functor LocSys `,n (X) 2 Fun Afd op k , Grpd is representable by a geometric k-analytic stack. It admits a smooth atlas by a k-analytic space LocSys framed `,n (X) ! LocSys `,n (X), where LocSys framed `,n (X) parametrizes framed étale `-adic local systems on X. Furthermore, LocSys `,n (X) admits a natural derived enhancement. Given ⇢ 2 LocSys `,n (X)(Q `), the analytic cotangent complex of LocSys `,n (X) at ⇢ is naturally equivalent to

L an

LocSys `,n (X),⇢ ' C ⇤ ét X, Ad(⇢)

_ [ 1] 2 Mod Q `,
where Ad(⇢) := ⇢ ⌦ ⇢ _ denotes the adjoint representation of ⇢.

Another important result concerning the geometry of LocSys `,n (X) is the existence of a natural shifted symplectic structure on LocSys `,n (X): Theorem 1.2.1.2. Let X be a proper and smooth scheme of dimension d. The moduli k-analytic stack LocSys `,n (X) admits a canonical 2 2d-shifted symplectic structure, whose underlying 2 2d-form is induced by Poincaré duality for étale cohomology (with derived coefficients).

In order to endow LocSys `,n (X) with a derived structure we had to develop new techniques to address the existence of formal models for derived k-analytic spaces. More precisely, we generalized to derived setting a well known theorem of Raynaud stating that the category of k-analytic spaces can be obtained as a localization of the category of formal models over k , satisfying certain finiteness conditions: Theorem 1.2.1.3. Let dAn k denote the 1-category of derived k-analytic spaces and dfSch k the 1-category of (admissible) derived k -adic schemes. Then there exists a derived rigidification functor ( ) rig : dfSch k ! dAn k which coincides with the usual rigidification functor for ordinary k -adic schemes. Moreover, the derived rigidification functor induces an equivalence

( ) rig : dfSch k [S 1
] ! dAn k of 1-categories, where S denotes the class of generically strong admissible blow-ups in dfSch k .

The above theorem has found many different applications. We have already mentioned the construction of the derived structure on LocSys `,n (X). Other such examples concern descent results for derived 1-categories of almost perfect modules on X 2 dAn k . One important application of the theory of formal models for derived k-analytic spaces is the following theorem proved in a joint work with M. Porta: Theorem 1.2.1.4. Let X be a quasi-separated and proper k-analytic space. Then the derived Hilbert stack associated to X RHilb(X) 2 dSt An k , ⌧ ét is representable by a derived k-analytic stack.

We can also show the existence of dualizing sheaves for derived k-analytic spaces, which as far as the knowledge of the author is concerned it is an original result. A main ingredient in the proof of the existence of a shifted symplectic form on LocSys `,n (X) one needs a rigid k-analytic version of the algebraic HKR theorem, proved by B. Toën and G. Vezzosi. This is a joint work in progress with F. Petit and M. Porta.

Theorem 1.2.1.5. Let k be the field C of complex numbers or a non-archimedean field of characteristic 0 with a non-trivial valuation. Let X be a k-analytic space. Then there is an equivalence of derived k-analytic spaces X ⇥ X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] compatible with the canonical projection to X.

Motivations 1.3.1 Non-abelian Hodge Theory

Let X be a topological space. The moduli scheme of C-local systems on X, denoted LocSys C,n (X), has been studied extensively in classical algebraic geometry. It can be defined by means of a moduli functor LocSys C,n (X) framed : A↵ C ! Set given on objects by the formula

A 2 A↵ C 7 ! Hom grp ⇡ top 1 (X), GL n (A) 2 Set,
where Hom grp denotes the set of group homomorphisms. The moduli space LocSys C,n (X) framed admits a canonical action of the general linear group scheme GL n 2 Sch C via conjugation. We can form the corresponding universal categorical quotient, which we shall denote LocSys C,n (X). It is then possible to show via geometric invariant theory, that LocSys C,n (X) is representable by a scheme of finite type over C, [Sim94a,[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. By construction, LocSys C,n (X) parametrizes semisimple rank n group representations of the topological fundamental group ⇡ top 1 (X). It is a well known fact that there exists a natural bijection between the set of rank n representation of ⇡ top 1 (X)

⇢ : ⇡ top 1 (X) ! GL n (C)
and the set of isomorphism classes of rank n local systems on X. The latter set can be identified with the underlying set of the cohomological group H 1 X, GL n (C) . For this reason, one can interpret the moduli LocSys `,n (X) as a non-abelian analogue of singular cohomology H 1 X, C on X with C-coefficients. Suppose now that X is a complex smooth projective variety. One can canonically equip the set of C-points of X with the structure of a topological manifold. As a consequence, to X we can associate an homotopy type

X(C) 2 S.
Concretely, the topological manifold X(C) corresponds to the underlying topological space of the analytification X an of X. One often refers to X(C) 2 S as the Betti realization of X. We can thus consider the moduli scheme LocSys C,n (X) := LocSys C,n X(C) , which parametrizes rank n-local systems on the Betti realization X(C) of X 2 Sch C . Let us introduce two other main ingredients in non-abelian Hodge Theory: Definition 1.3.1.1. Let X be a projective and smooth variety over the field C of complex numbers. We define the moduli stack Conn C,n (X) : A↵ C ! S given on objects by the formula

A 2 A↵ C 7 ! Conn C,n (X)(A) 2 S
where Conn C,n (X)(A) denotes the space of rank n flat connections with A-coefficients over X. We can also consider the moduli stack Higgs(X) : A↵ C ! S parametrizing rank n Higgs bundles with A-coefficients on X, see [Sim94a,[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]p. 15]. Moreover, C. Simpson in his seminal work on non-abelian Hodge theory proved that the analytifications Conn an C,n (X) and Higgs an (X) are homeomorphic. This last result can be interpreted as a non-abelian analogue of the degeneration of the Hodge to de Rham spectral sequence. The above definition defines a more general object than the universal categorical quotient of LocSys framed C,n (X). Indeed, the latter parametrizes semisimple representations of the fundamental group ⇡ top 1 (X) whereas the former parametrizes the space of all group representations of ⇡ top 1 (X). Moreover, the formula (1.3.1.1) can be easily generalized to the derived setting. We can define the derived enhancement of LocSys C,n (X) as the derived mapping stack LocSys `,n (X) := Map dSt X(C), BGL n ( ) 2 dSt dA↵ C , ⌧ ét , P sm , where dSt dA↵ C , ⌧ ét , P sm denotes the 1-category of geometric stacks with respect to the algebraic geometric context dA↵ C , ⌧ ét , P sm . Similarly, both the moduli Conn C,n (X) and Higgs(X) can be upgraded to derived C-Artin stacks via the equivalences of stacks Conn C,n (X) ' Map dSt X dR , BGL n ( ) , Higgs(X) ' Map dSt X Dolb , BGL n ( ) .

We refer the reader to [Por17b,[START_REF] Bhatt | The pro-étale topology for schemes[END_REF] for the definition of the notions X dR and X Dolb .

Theorem 1.3.1.3. [Por17b, Theorem 6.11] Let X be a projective smooth complex variety. The Riemann-Hilbert correspondence induces an equivalence

This can be regarded as a vast generalization of the usual comparison isomorphism between de Rham and Betti cohomologies for smooth and proper varieties over C. At the heart of the proof of the above equivalence lies the Riemann-Hilbert correspondence between the categories of rank n local systems on X and rank n flat connexions on X. Question 1.3.1.4. What can be said for smooth and proper schemes over a algebraically closed fields of characteristic p > 0. More precisely, is there any non-abelian analogues of p-adic Hodge theory?

One main ingredient in our previous discussion was the existence of the homotopy type X(C) 2 S of C-points of X. Unfortunaly, in positive characteristic the sole analogue of X(C) is the étale homotopy type of X, which we shall denote Sh ét (X). In this case, Sh ét (X) is not a homotopy type but instead Sh ét (X) 2 Pro(S fc ), i.e. Sh ét (X) is a profinite space or profinite homotopy type.

It has long been understood that there are deficiencies with studying study continuous representations

⇢ : ⇡ ét 1 (X) ! GL n (C)
The main reason is the fact that any such ⇢ must necessarily factor via a finite subgroup of GL n (C). Nonetheless, in positive characteristic, one usually studies rank n `-adic étale lisse sheaves on X. Therefore, the objects of our interest correspond to continuous representations

⇢ : ⇡ ét 1 (X) ! GL n (Q `). (1.3.1.2)
Ultimately, one is also interested in studying continuous representations of homotopy types

⇢ : Sh ét (X) ! BGL n (Q `)
However, we do not have a clue of what shall mean a continuous representations of homotopy types. We present the reader with a list of properties that we would like such continuous representations satisfy:

(i) Let A 2 CAlg Q `be a derived Q `-algebra. There should exist a space of continuous representations

⇢ : Sh ét (X) ! BGL n (A). (1.3.1.3)
Moreover, if A ' ⇡ 0 (A). There should be a natural equivalence between the space of such objects (1.3.1.3) and the space of continuous representations ⇢ : B⇡ ét 1 (X) ! BGL n (A).

(ii) For each continuous representation ⇢ : Sh ét (X) ! BGL n (A) one should be able to a find a commutative diagram

U BGL n (B) Sh ét (X) BGL n (A) ⇢ U ⇢
where B 2 CAlg Z `is of no `-torsion and one has furthermore an equivalence

B ⌦ Z `Q`' A 2 CAlg Q `.
Moreover U 2 Pro(S fc ) /Sh ét (X) denotes a profinite space such that one has a fiber sequence

Y ! U ! Sh ét (X),
with Y 2 S fc is a finite constructible space.

(iii) Let A 2 CAlg Q `be a derived Q `-algebra. A continuous representation

⇢ : Sh ét (X) ! BGL n (A)
should induce, via extension of scalars, a continuous representation

⇢ : Sh ét (X) ! BGL n (B ⌦ Z `F`)
where B 2 CAlg Z `is as in (ii). We require furthermore that such continuous ⇢ factor through a finite quotient X i 2 S fc of Sh ét (X) 2 Pro(S fc ).

In order to facilitate our exposition we adopt the following convention for the mapping space of continuous representations, which we have not yet defined: Notation 1.3.1.5. Given A 2 CAlg Q `we shall denote by

Map

cont Sh ét (X), BGL n (A) 2 S the space of continuous representations ⇢ : Sh ét (X) ! BGL n (A).

Suppose we have a reasonable definition of a notion continuous representation as in (1.3.1.2). Then we can define the moduli functor LocSys `,n (X) : dAfd op Q `! S as given by

Z := Z, O Z 2 dAfd op Q `7 ! Map cont ⇣ Sh ét (X), (Z) ⌘ 2 S
where (Z) denotes the global sections ring of Z, defined as

⇡ ⇤ (O Z 2 CAlg Q `.
The derived moduli stack LocSys `,n (X) is an `-adic analogue of the moduli of rank n complex local systems for complex varieties, LocSys C,n (X).

Remark 1.3.1.6. It would be desirable that certain results of p-adic Hodge for proper smooth schemes over Q p admit non-abelian Hodge theoretical analogues. And in such case, the moduli LocSys p,n (X) should play a role. However, there are serious obstructions for a complete p-adic analogue as we shall see in later sections.

Deligne Comptage

In his seminal work [Dri81] V. Drinfeld proved a counting formula for rank 2 `-adic lisse sheaves on a smooth and proper curve X, up to -torsion. Drinfeld's formula uses in an essential way his work on the Langlands correspondence, [Dri80]. More recently, in the work of [Yu18] such formula was generalized to higher ranks and to the open case by considering fixed monodromy at infinity.

In [Del15], P. Deligne conjectured that the counting problem of rank n `-adic étale lisse sheaves on a smooth variety X over a finite field F q could be stated equivalently as a Grothendieck-Lefschetz trace formula on a suitable moduli space of `-adic local systems. In order to understand his assertion recall that Galois descent induces a bijection between the set of `-adic lisse sheaves on X and the set of `-adic lisse sheaves over the base change X := X ⇥ Fq F q which admit a Z `-lattice stable under pullback along the geometric Frobenius endomorphism of X, Frob : X ! X.

More precisely, given F a Q `-adic sheaf on X we can consider it as an `-adic sheaf on X, via pullback along the canonical map f : X ! X. Moreover, the `-adic sheaf F satisfies Frob ⇤ F ' F in the category of `-adic lisse sheaves on X. The crucial observation of P. Deligne is the fact that Drinfeld's formula is reminiscent of a Grothendieck-Lefschetz trace formula with respect to the pair LocSys `,n (X), F , where LocSys `,n (X) should correspond to the moduli of `-adic continuous representations of the étale fundamental group ⇡ ét 1 (X) and F : LocSys `,n (X) ! LocSys `,n (X) denotes the endomorphism of LocSys `,n (X) given on objects by the formula F 2 LocSys `,n (X) 7 ! Frob ⇤ F 2 LocSys `,n (X).

One would then like to confirm that F admits a finite number of fixed points and such number could be computed by means of a trace formula for LocSys `,n (X), F . However, in [Del15] the author does not give any hint on how to construct LocSys `,n (X) except for certain complex analogies, inspired mainly by non-abelian Hodge theory. Therefore, in order to prove Deligne's conjecture one would have to show the following statements:

(i) There exists a natural candidate LocSys `,n (X) for the moduli of `-adic continuous representations of ⇡ ét 1 (X). Moreover, such candidate should have sufficiently geometric properties, such as being representable by an algebraic or Q `-analytic stack. In particular, one should have a complete understanding of its deformation theory around a closed point

⇢ : ⇡ ét 1 (X) ! GL n (Q `).
(ii) There exists a reasonable cohomological theory on LocSys `,n (X), which we shall denote by

C ⇤ • (LocSys `,n (X)).
(iii) The automorphism F : LocSys `,n (X) ! LocSys `,n (X) admits a finite set of fixed points and a trace formula holds with respect to the triplet LocSys `,n (X), C ⇤ • (LocSys `,n (X)), F .

Moreover, it would be interesting if one is able to prove the above conjectural statements by purely geometric means without need to pass to the automorphic setting.

In this thesis we will answer positively to (i) and (ii). We construct LocSys `,n (X) directly as a Q `-analytic stack, whose proof follows roughly the same lines as in the complex case. We show thereafter that it is possible to enhance LocSys `,n (X) with a natural derived structure. Such derived structure allow us to consider derived de Rham cohomology on LocSys `,n (X), C ⇤ dR LocSys `,n (X) . However, we will show that LocSys `,n (X) has too many connected components. Indeed, the moduli stack LocSys `,n (X) admits one connected component for each residual representation ⇢ : ⇡ ét 1 (X) ! GL n (F `r . Therefore, our results cannot applied directly to show Deligne's conjecture, however we will indicate some possible future avenues in later chapters.

Theory of formal models for ordinary k-analytic spaces

Let us illustrate a possible route to solve the question posed in Theorem 2.1.0.15. We start by recall the main results concerning the existence of formal models for k-analytic spaces: Theorem 1.3.3.1 ( §8 [Bos05]). Let X 2 An k denote a quasi-paracompact and quasi-separated k-analytic space. Then there exists an admissible formal k -scheme X 2 fSch k such that one has an equivalence X rig ' X, in the category An k .

Theorem 1.3.3.1 can be stated equivalently as the essential image of the functor ( ) rig coincides with the full subcategory of An k spanned by quasi-paracompact and quasi-separated k-analytic spaces. In particular, Theorem 1.3.3.1 implies that whenever X = Sp(A) is k-affinoid, we can find an affine formal model of the form Spf(A 0 ) 2 fSch k such that A 0 is an admissible k -algebra. In other words, A 0 is topologically of finite presentation and m-torsion free, where m denotes the maximal ideal of k .

Let us illustrate how a derived analogue of Theorem 1.3.3.1 is helpful to treat the question of Theorem 2.1.0.15. Suppose that k is a finite extension of Q `. For every A 2 AnRing sm k we can find a formal model A 0 2 CAlg sm k , i.e. there exists A 0 verifying the following conditions:

(i) The ordinary commutative ring ⇡ 0 (A 0 ) admits an adic topology compatible with the one on k ;

(ii) One has an equivalence

A 0 ⌦ k k ' A alg in the 1-category CAlg k .
In particular, thanks to [Lur16,Remark 8.1.2.4] the derived k -algebra A 0 can be realized as an inverse limit A 0 ' lim n 0 A 0,n in the 1-category CAlg k , where A 0,n denotes the pushout diagram

A 0 [t] A 0 A 0 A 0,n t7 !`n t7 !0
in the 1-category CAlg k . In this case, the classifying space BGL n (A 0 ) 2 S can be realized as an object in the 1-category Pro(S fc ) of profinite spaces. Namely, one can consider the object

BGL n (A 0 ) := {BGL n (A 0,m )} m 2 Pro(S fc
).

(1.3.3.1)

Moreover, as the transition maps in (1.3.3.1) are compatible with the group structures for different m it follows that BGL n (A 0 ) 2 Pro(S fc ) can be promoted to an object in Mon grp E1 Pro(S) .

Remark 1.3.3.2. Suppose that one is provided with a functorial assignement

A 0 2 CAlg ad k 7 ! BGL n (A 0 ) 2 Mon grp E1 (S fc ),
where CAlg ad k denotes the 1-category of derived adic k -algebras. In this case, one could expect to define a functor

F : AnRing sm k ! S
given on objects by the formula

A ! k 2 AnRing sm k 7 ! Map cont B⇡ ét 1 (X), BAut(A n ) Map cont (B⇡ ét 1 (X),BAut(k n )) {⇢} 2 S (1.3.3.2)
where Map cont B⇡ ét 1 (X), BAut(A n ) denotes the colimit

Map cont B⇡ ét 1 (X), BAut(A n ) := colim A0 Map
Mon E 1 (Pro(S fc )) B⇡ ét 1 (X), BAut(A n 0 ) .

over A 0 , varying through the 1-category of formal models for A 2 AnRing sm k . The formula displayed in (1.3.3.2) is already a good approximation of a formal moduli problem classifying continuous deformations of

⇢ : ⇡ ét 1 (X) ! GL n (k).
However, there are still certain issues which render the formula (1.3.3.2) problematic. First, the étale homotopy group ⇡ ét 1 (X) does not classify `-adic lisse sheaves on X ét with derived coefficients. One should instead replace the profinite group ⇡ ét 1 (X) with the étale homotopy type of X, Sh ét (X). Fortunately, the formula in (1.3.3.2) is sufficiently general so that we can replace B⇡ ét 1 (X) 2 Pro(S fc ) with Sh ét (X) 2 Pro(S fc ) or any other profinite space without concern.

Secondly and more importantly, formula (1.3.3.2) does not classify all deformation of ⇢, instead only those continuous deformation of ⇢ which are power bounded. However, in general, there are many continuous deformations of ⇢ which do not factor through power bounded matrices, even in the ordinary case. Nonetheless, (1.3.3.2) is simple enough to hint us a reasonable notion of continuity in the derived setting. We will delve this question in further detail in §4. Also, (1.3.3.2) motivates an analogous statement of Theorem 1.3.3.1 in the derived setting.

At this point, the avid reader might object that the formula displayed in (1.3.3.2) is not necessarily functorial on A 2 AnRing sm k . In order to show functoriality of (1.3.3.2) one needs to generalize the following fundamental result due to Raynaud: Theorem 1.3.3.3 (Theorem 8.4.3 [Bos05]). The ordinary rigidification functor ( ) rig : fSch k ! An k factors through the localization fSch k [S 1 ], where S denotes the class of admissible ups on fSch k . Moreover, ( ) rig induces an equivalence of catgories

fSch k [S 1
] ' An 0 k where An 0 k denotes the full subcategory of An k spanned by quasi-paracompact and quasi-separated k-analytic spaces.

If we are able to generalize ?? 1 to the derived setting then we would be able, via a formal reasoning, to show that the assignment in (1.3.3.2) is functorial. The generalization of both ?? 1 and Theorem 1.3.3.1 are now proven facts which make part of the current thesis project which we detail in this introduction. which associates to a formal k -scheme X 2 fSch k its rigidification X rig 2 An k . Henceforth, in order to state derived analogues of both Theorem 1.3.3.1 and ?? 1 one would need the following derived analogues:

(i) An 1-category of derived k -adic schemes, dfSch k which have been introduced in [Lur16, §8];

(ii) An 1-category of derived k-analytic spaces, dAn k , introduced in [PY16a];

(iii) A derived rigidification functor ( ) rig : dfSch k ! dAn k which restricts to (1.4.1.1) on ordinary k -adic schemes.

Even though items (i) and (ii) have been treated extensively in the literature, it is not clear how to directly define a derived rigidification functor ( ) rig : dfSch k ! dAn k .

A major obstable results from the fact that the 1-category dfSch k is defined in [Lur16,[START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF] based on the theory of locally ringed 1-topoi. More precisely, derived formal Deligne-Mumford stacks correspond to (X, O) where X is an 1-topos and O 2 CAlg k (X) is a local derived k -algebra on X such that the ordinary commutative ring sheaf ⇡ 0 O) 2 CAlg k (X) is equipped with an adic topology compatible with the adic topology on k . On the other hand, the 1-category dAn k is defined in terms of T an (k)-structured spaces. Unfortunately, no direct comparison exists between adic locally ringed 1-topoi and T an (k)-structured 1-topoi. A possible way to deal with this difficulty is to redefine derived k -adic geometry via a structured 1-topoi approach, where we consider structured 1-topoi with respect to a suitable k -adic pregeometry, T ad (k ). After adopting such a viewpoint, one is equipped for free with a transformation of pregeometries

( ) rig : T ad (k ) ! T an (k)
induced from the classical rigidification functor. Moreover, by a formal reasoning one can prove that such transformation of pregeometries provide us with a candidate for a derived rigidification functor dfSch k ! dAn k . For this reason, we adopt the structured spaces point of view for derived k -adic geometry.

Definition 1.4.1.1. Let T ad (k ) denote the full subcategory of fSch k spanned by those ordinary affine formal k -schemes Spf(A) such that Spf(A) is étale over some affine n-space, A n k . We equipp T ad (k ) with the étale topology. The class of admissible morphims on T ad (k ) is defined as the class of étale morphisms in T ad (k ). Definition 1.4.1.2. We defined the 1-category of T ad (k )-structured spaces as the 1-category of couples (X, O) such that X 2 Top R is an 1-topos and O : T ad (k ) ! X is a local T ad (k )-structure on X. Notation 1.4.1.3. Let X be an 1-topos. The 1-category of local structures on X is denoted by fCAlg k (X).

As in the setting of derived k-analytic geometry, one has a well defined, up to contractible indeterminacy, underlying algebra functor ( ) alg : fCAlg k (X) ! CAlg k (X), given on objects by the formula

O 2 fCAlg k (X) 7 ! O(A 1 k ) 2 CAlg k (X).
Moreover, this functor can be promoted to a functor whose target consists of derived k -algebras on X:

Lemma 1.4. 1.4. The underlying algebra functor ( ) alg : fCAlg k (X) ! CAlg k (X) can be naturally promoted to a well defined, up to contractible indeterminacy, functor

( ) ad : fCAlg k (X) ! CAlg k (X),
which is given on objects by the formula

O 2 fCAlg k (X) 7 ! O(A 1 k ) 2 CAlg ad k (X)
where O(A 1 k ) 2 CAlg ad k (X) is equipped with natural adic topology, induced by the sequence of ideals {I n } n of ⇡ 0 (O alg ) which correspond to the kernel of the canonical ring homomorphisms

⇡ 0 (O alg ) ! ⇡ 0 (O alg ⌦ k k n )
for each n 1.

The following fundamental result implies that specifying local T ad (k )-structures on X is roughly equivalent as specifying a derived k -adic locally ring on X: Theorem 1.4.1.5. Let X be an 1-topos with enough geometric points. Then the functor

( ) ad : fCAlg k (X) ! CAlg ad k (X)
induces an equivalence of the 1-categories of topologically almost of finite presentation objects ( ) ad : fCAlg k (X) taft ! CAlg ad k (X) taft .

Remark 1.4.1.6. Theorem 1.4.1.5 can be interpreted as a rectification type statement. Indeed, specifying a local T ad (k )-structure on X consists in specifying a functor O : T ad (k ) ! X satisfying the admissibility conditions of Theorem 2.1.0.2. A priori, one would expect that the required amount of higher coherence data for T ad (k )structures should be considerably more complex than the higher coherence data specifying the underlying algebra object O alg . Theorem 1.4.1.5 imply that the difference can be measured by the given of an adic topology on the ordinary ring object ⇡ 0 (O alg ). Moreover, morphisms between local structures correspond to morphisms of functors T ad (k ) ! X which satisfy the local condition in Theorem 2.1.0.2. Theorem 1.4.1.5 imply that these amount to the same higher coherent data as specifying continuous adic morphisms between derived commutative k -algebras on X. The latter morphisms correspond to morphisms in the 1-category CAlg k (X) satisfying a continuity condition which can be verified directly at the level of ⇡ 0 . Definition 1.4.1.7. Let X := (X, O) 2 Top R (T ad (k )) be a T ad (k )-structured 1-topos. We say that X is a derived k -adic Deligne-Mumford stack if the k -adic locally ringed 1-topos X ad := (X, O ad

) is a derived formal Deligne-Mumford stack in the sense of [Lur16,Definition 8.1.3.1].

As in [Lur16,[START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF] we can define a Spf-construction which will prove to be very useful for us: Proposition 1.4.1.8. Let Spf : CAlg ad k op ! Top R (T ad (k )) be the Spf-construction. Then Spf is fully faithful and its essential image consists of those pairs X, O) such that X, O ad is equivalent to an affine derived k -adic Deligne-Mumford stack as in [Lur16,[START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF].

Moreover, as in the derived k-analytic setting we can show that the category of ordinary k -adic Deligne-Mumford stacks can be realized as a full subcategory of dfSch k via the following construction: Construction 1.4.1.9. Let X 2 fDM k denote an ordinary k -adic Deligne-Mumford stack. Consider the étale site X ét and the hypercompletion X := Shv ét (X) of the 1-localic 1-topos of étale sheaves on X. We can define a T ad (k )-structure, O : T ad (k ) ! X on X as follows: given V 2 T ad (k ) we associate it the sheaf O(V ) given on objects by the formula U 2 X ét 7 ! Map fDM k (U, V ) . Moreover, the CAlg k (X)-sheaf O(A 1 k ) corresponds to the usual sheaf of k -adic global section on X. Fortunately, we are now in position to define a rigidification functor: consider the transformation of pregeometries ( ) rig : T ad (k ) ! T an (k) given by restricting the usual rigidification functor to the category T ad (k ). Precomposition along ( ) rig : T ad (k ) ! T an (k) induces a functor The following proposition implies that the restriction of ( ) rig : dfDM k ! dAn k to the category of ordinary k -adic schemes coincide with the usual rigidification functor.

Proposition 1.4.1.11. Let X 2 dfDM k be a derived k -adic Deligne-Mumford stack which is equivalent to an ordinary k -adic scheme. Then X rig coincides with usual rigidification functor for ordinary k -adic schemes.

The following two results are direct generalizations of both Theorem 1.3.3.1 and ?? 1 to the derived setting: Theorem 1.4.1.12. Let X 2 dAn k be a quasi-paracompact and quasi-separated derived k-analytic space. Then there exists X 2 dfDM k such that one has an equivalence X rig ' X in the 1-category dAn k . ] of dfSch k , where S denotes the class of admissible blow ups and generally strong morphisms. Moreover, it induces an equivalence of 1-categories

dfSch k [S 1
] ' dAn 0 k where dAn 0 k denotes the full subcategory of dAn k spanned by quasi-paracompact and quasi-separated derived k-analytic spaces.

As a corollary we obtain: Corollary 1.4.1.14. Let f : X ! Y be a morphism in the 1-category dAn k . Then we can find a morphism f : X ! Y in dfDM k such that one has an equivalence (f) rig ' f in the 1-category of morphisms dAn 1 k . The results on the existence of derived formal models, namely Theorem 6.2.3.15 and Theorem 3.4.4.10 have found applications so far. In the next chapter we cover certain of these applications:

Derived k-analytic Hilbert moduli stack

The contents of this chapter were proven in a joint work with M. Porta.

Let X be a proper variety. We can associate to X its Hilbert scheme, denoted Hilb(X). Hilb(X) is defined via its functor of points. Roughly, Hilb(X) parametrizes closed subschemes of X flat over the base. More precisely, Hilb(X) represents the functor Hilb(X) : Sch k ! Set, which associates to a scheme S 2 Sch k the set of closed subschemes of X ⇥ S which are flat over S. When X = P n we recover the usual Hilbert scheme Hilb(n) parametrizing closed subschemes of the projective nspace P n . The moduli scheme Hilb(X) plays an important role in many representability statements, including an important role in geometric invariant theory. It would thus be desirable to extend the construction of Hilb(X) to the k-analytic setting.

This was achieved in [CG16], in the ordinary setting. The authors prove that given a separated k-analytic space X, one can associate it a k-analytic Hilbert space, Hilb an (X) which parametrizes flat families of closed subschemes of X.

However, the requirement of flatness in the above definition is restrictive, both in the algebraic and k-analytic settings. One would like to not only parametrize flat families of closed subschemes of X but all families of closed subschemes of X. A possible way to deal with this issue is to consider a natural derived enhancement of Hilb(X), namely the derived Hilbert stack RHilb(X). The representability of RHilb(X) as a geometric stack has been establish in the context of derived algebraic geometry via the Lurie-Artin representability theorem [Lur12a, Theorem 3.2.1].

[PY17a, Theorem 7.1] provides a derived k-analytic analogue of Lurie-Artin representability theorem. Therefore, one could hope that the derived k-analytic Hilbert space RHilb an (X) could be shown to be representable directly using [PY17a,Theorem 7.1].

However, via this approach we are allowed to prove the existence of an analytic cotangent complex of RHilb an (X) only at points f : S ! RHilb an (X) corresponding to families of closed subschemes of j : Z ,! X ⇥ S which are of finite presentation, in the derived setting. However, not all points of RHilb an (X) satisfy this condition, we are typically interested with families which are almost of finite presentation. Definition 1.4.2.1. We denote by dSt dAfd k , ⌧ ét , P sm the 1-category of geometric stacks with respect to the derived k-analytic geometric context. We refer to objects of dSt dAfd k , ⌧ ét , P sm simply as derived k-analytic stacks.

Using the techniques developped in [Ant18b] together with [PY17a,Theorem 7.1] we are then able to prove the following main result: Theorem 1.4.2.2. Let X be a separated ordinary k-analytic space. The derived k-analytic Hilbert stack RHilb an (X) is representable by a derived k-analytic stack.

In order to prove Theorem 1.4.2.2 the authors had to generalize certain results related to the existence of formal models for certain classes of morphisms between derived k-analytic spaces. Namely, we generalized to the derived setting a classical result of Bosch-Lutkebohmer concerning liftings of flat morphisms f : X ! Y to flat morphisms of formal models. The main difficulty proving Theorem 1.4.2.3 was to verify the strong condition for derived morphisms at the formal level. More precisely, the main obstruction to apply Theorem 3.4.4.10 directly was to verify the condition

⇡ i f ⇤ O Y ' ⇡ i O X
for i > 0, where f : X ! Y is a generic formal model for f : X ! Y . This was accomplished by a reasoning at the level of Postnikov towers plus proving refined results concerning liftings of formal models for almost coherent modules on X. At the heart of such liftings results is the following descent result: (X) Coh b (X) /F spanned by formal models for F is filtered.

Proposition 1.4.2.6. Let X 2 dAn k be a derived k-analytic space. Let f : F ! G be a morphism of almost coherent modules in the 1-category Coh + (X). Then for every formal model X 2 dfDM k of X and a choice of formal models F 0 and G 0 for F and G, respectively, there exists a sufficiently large n 0 such that the multiplication t n f : F ! G admits a formal model of the form

g : F 0 ! G 0 in the 1-category Coh + (X).
As an application of Theorem 1.4.2.3 we obtain the following:

Proposition 1.4.2.7. Let f : X ! Y be a flat and proper morphism of derived k-analytic spaces. Then we have:

(i) The functor f ⇤ : Coh + (Y ) ! Coh + (X) admits a left adjoint f + : Coh + (X) ! Coh + (Y ).
(ii) Let f : X ! Y be a formal model for f , whose existence is guaranteed by Theorem 1.4.1.14. Define ! X/Y := ! rig X/Y , the analytic dualizing sheaf. Then given F 2 Coh + (X) we have a canonical equivalence

f + (F) ' f ⇤ (F ⌦ ! X/Y ) in the 1-category Coh + (Y ). (iii) The functor f ! : Coh + (Y ) ! Coh + (Y )
given on objects by the formula

F 2 Coh + (Y ) 7 ! f ! (F) := f ⇤ (F ⌦ ! X/Y ) 2 Coh + (X)
is a right adjoint for the functor

f ⇤ : Coh + (X) ! Coh + (Y ).
1.5 Moduli of continuous `-adic representations of a profinite group

Moduli of continuous `-representations

Part of the present thesis was to study at length the moduli functor LocSys `,n (X) parametrizing continuous `-adic representations of ⇡ ét 1 (X), where X is a proper and smooth variety over an algebraically closed field of positive characteristic p > 0.

In this context, the formal moduli problem considered in Theorem 2.1.0.15 should correspond to the formal neighborhood of the moduli LocSys `,n (X), described in Section 1.3.2. Furthermore, Theorem 3.4.4.10 allow us to define general continuous representations

⇢ : Sh ét (X) ! BGL n (Z) ,
where Z 2 dAfd Q `is not necessarily the spectrum of an Artinian derived Q `-algebra. Notice that when Z is discrete we can describe and study the moduli problem associated to LocSys `,n (X) directly. Consider the following moduli functor LocSys framed `,n (X) : Afd op k ! Set given on objects via the formula

A 2 Afd op k 7 ! Hom cont ⇡ ét 1 (X), BGL n (A) 2 Set,
where Hom cont ⇡ ét 1 (X), BGL n (A) denotes the set of continuous group homomorphisms ⇡ ét 1 (X) ! GL n (A). Here we consider GL n (A) as a topological group via the canonical topology on A induced by a choice of norm compatible with the non-trivial valuation on k. We have the following fundamental result: Theorem 1.5.1.1. Let X be a smooth and proper scheme over an algebraically closed field. The functor

LocSys framed `,n (X) : Afd op k ! Set
defined above is representable by a strict k-analytic space.

The strategy of the proof of Theorem 1.5.1.1 follows closely the scheme of Simpson's proof the representability of the moduli of discrete group homomorphisms for a discrete group G, see [Sim94a,[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. However, the continuous case is more involved as both the topologies on ⇡ ét 1 (X) and on GL n (A) differ. More precisely, ⇡ ét 1 (X) is a profinite group whereas the topology on GL n (A) is far from being profinite. Nonetheless, every formal model

A 0 over k for A provides an open subgroup GL n (A 0 ) ✓ GL n (A)
which is a pro-group, i.e. the topology on GL n (A 0 ) is induced via the canonical isomorphism of groups

GL n (A 0 ) ' lim m 0 GL n (A 0,m ).
In this case, every continuous group homomorphism ⇢ :

⇡ ét 1 (X) ! GL n (A) admits an open subgroup U  ⇡ ét 1 (X) such that ⇢ |U : U ! GL n (A) factors through the inclusion GL n (A 0 ) ✓ GL n (A).
Fortunately, the study of group homomorphisms

⇢ |U : U ! GL n (A 0 ) (1.5.1.1)
is easier than our original problem. Therefore, it is useful to study the moduli functor F ⇡ ét 1 (X) : CAlg ~,ad k ! Set parametrizing continuous group homomorphisms

⇢ : U ! GL n (A 0 ), A 0 2 CAlg ~,ad k .
Nevertheless, the topology on GL n (A 0 ) is almost never profinite, except when A 0 is a finite extension of k , therefore some care is needed when describing the above functor by means of algebraic data. Even though F ⇡ ét 1 (X) is not representable, Theorem 1.5.1.1 implies that k-analytic analogue LocSys framed `,n (X) is so.

Remark 1.5.1.2. The k-analytic space LocSys framed `,n (X) admits a natural action of the analytification of the general linear group scheme, GL an n . This action can be described via the morphism of k-analytic spaces

GL an n ⇥ LocSys framed `,n (X) ! LocSys framed `,n (X), (1.5.1.2)
defined by the formula

(g, ⇢) 2 GL an n (A) ⇥ LocSys framed `,n (X)(A) 7 ! g⇢g 1 2 LocSys framed `,n (X)(A) .
One would like to define LocSys `,n (X) as the quotient k-analytic space obtained as the quotient of LocSys framed `,n (X) by the conjugation action of GL an n described in Theorem 1.5.1.2. However, in the k-analytic setting there is no solid theory of geometric invariant theory as opposed to the context of algebraic geometry. For this reason we prefer to adopt the language of k-analytic stacks for a reasonable definition of LocSys `,n (X). Definition 1.5.1.3. We denote by LocSys `,n (X) the geometric realization of the simplicial objects

• • • GL an n 2 ⇥ LocSys framed `,n (X) GL an n ⇥ LocSys framed `,n (X) LocSys framed `,n (X) (1.5.1.
3) computed in the 1-category of pre-sheaves on Afd k , denoted PShv Afd k := Fun (Afd op k , S). Remark 1.5.1.4. The moduli functor LocSys `,n (X) : Afd op k ! S satisfies descent with respect to the étale topology on (Afd k , ⌧ ét ). Furthermore, it follows by construction that given A 2 Afd k the space LocSys `,n (X)(A) 2 S is equivalent to the 1-groupoid of continuous representations ⇢ : ⇡ ét 1 (X) ! GL n (A). Therefore, LocSys `,n (X) parametrizes continuous `-adic group representations of ⇡ ét 1 (X) or equivalently rank n pro-étale local systems on X ét as the following result illustrates: Proposition 1.5.1.5. Let X be a smooth scheme over an algebraically closed field. Then one has a natural equivalence of spaces

LocSys `,n (X)(A) ' Loc n,pro-ét (X)(A),
where the right hand side denotes the 1-groupoid of rank n pro-étale A-linear local systems on X.

Theorem 1.5.1.1 entails through a formal reasoning the following main result: Theorem 1.5.1.6. The moduli functor LocSys `,n (X) : Afd op k ! S is representable by a geometric stack with respect to the geometric context Afd k , ⌧ ét , P sm . Notation 1.5.1.7. We refer to geometric stacks with respect with the geometric context Afd k , ⌧ ét , P sm simply as k-analytic stacks.

Theorem 1.5.1.6 provides a positive answer to condition (i) in §1.5. However, we have a very few understanding of the geometry of LocSys `,n (X) and a reasonable cohomology theory for LocSys `,n (X) is still lacking. We will try to amend these questions by constructing a natural derived structure on LocSys `,n (X).

Derived enhancement of LocSys `,n (X)

Let X be a geometrically connected proper and smooth scheme over an algebraically closed field. The moduli stack LocSys `,n parametrizes varying families of rank n `-adic pro-étale local systems on X. Moreover, Theorem 1.5.1.6 states that LocSys `,n (X) is representable by a k-analytic stack. In this section we will further attempt to answer the following question: Question 1.5.2.1. Does LocSys `,n (X) admits an analytic cotangent complex which classifies deformations of pro-étale local systems? And if so, can we compute it explicitly? Theorem 1.5.2.1 is basically a question on the existence of a canonical derived structure on LocSys `,n (X). In order to attempt to answer to Theorem 1.5.2.1 one needs to allow derived coefficients in the definition of LocSys `,n (X). More precisely, we nedd to extend the functor LocSys `,n (X) to a functor LocSys `,n (X) : dAfd op k ! S, so one needs a reasonable defintion of continuous representations with derived coefficients. This questions has already been partially dealt in Theorem 2.1.0.15. However, to fully answer this question one needs to make a considerable technical detour on the theory of enriched 1-categories. We start by observing that the étale fundamental group is too poor, in general, to classify `-adic lisse sheaves on X ét with derived coefficients. One should consider instead the étale homotopy type Sh ét (X) of X and parametrize continuous `-adic representations of Sh ét (X). Let Z 2 dAfd k , thanks to Theorem 3.4.4.10 there exists a formal model Spf (A 0 ) for Z. One could try to define continuous representations of homotopy types

⇢ : Sh ét (X) ! BGL n (Z)
as in Theorem 2.1.0.15 by considering the colimit over all such formal models for Z, i.e. by defining

Map cont Sh ét (X), BGL n (Z) := colim A0 Map Mon E 1 (Pro(S)) Sh ét (X), BGL n (A 0 ) 2 S. (1.5.2.1)
By considering BGL n (A 0 ) as a group-like pro-object in the 1-category S via the equivalence

A 0 ' lim m 0 A 0,m
in the 1-category CAlg k . Theorem 3.4.4.10 implies that such an association is functorial in Z 2 dAfd k . Even though formula (1.5.2.1) is a good approximation for the space of continuous `-adic representations of Sh ét (X) it is certainly not a correct definition. In order to give a more reasonable construction one need to make a technical detour on the theory of Ind(Pro(S))-enriched 1-categories. Construction 1.5.2.2. Let Z 2 Afd k be a derived k-affinoid space and consider the 1-category Perf(Z) of perfect complexes on Z. Fix a formal model Spf(A 0 ) 2 dfSch k for Z. The 1-categories Perf(Z) and Perf(A 0 ) are related. The derived rigidification theorem introduced in §3 induces a rigidification functor at the level of 1-categories of perfect modules

( ) rig : Perf(A 0 ) ! Perf(Z).
Moreover, as a first approximation we can think of Perf(Z) as the idempotent completion of the Verdier quotient Perf(A 0 )/Perf nil (A 0 ) where Perf nil (A 0 ) ✓ Perf(A 0 ) denotes the full subcategory spanned by m-torsion perfect A 0 -modules. In other words, Perf(Z) is roughly equivalent to the Verdier quotient

Perf(A 0 )/Perf nil (A 0 ) computed in the 1-category Cat st,id-comp 1 .
The crucial observation is that the 1-category Perf(A 0 ) is naturally enriched over the 1-category Pro Sp ` of pro-objects on the 1-category of `-nilpotent spectra, Sp `:= Sp ⌦ S S/`, where S 2 Sp denotes the sphere spectrum. By a formal argument, detailed in [Ant17a], we can then consider Perf(Z) as an 1-category naturally enriched over the 1-category of ind-pro-spaces, Ind Pro S . Moreover, the formula

Z 2 Afd op k 7 ! Perf(Z) 2 ECat 1 , is functorial in Z 2 Afd op k . Furthermore, there is a canonical inclusion functor Pro S fc ,! Pro S ,! ECat 1 ,
where ECat 1 denotes the 1-category of Ind(Pro(S))-enriched 1-categories. Therefore, it makes sense to consider the 1-category of enriched functors

Fun ECat1 Sh ét (X), Perf(Z) 2 Cat 1 . (1.5.2.2)
Moreover, the association displyed in (1.5.2.2) is functorial in Z. We can thus define a functor

Fun ECat1 Sh ét (X), Perf( ) : dAfd op k ! Cat 1
given on objects by the formula

Z 2 Afd op k 7 ! Fun ECat1 Sh ét (X), Perf(Z) 2 ECat 1 .
As X is geometrically connected, the profinite space Sh ét (X) 2 Pro(S fc ) is connected. For this reason, the mapping space

Map

Pro(S fc ) ⇤, Sh ét (X) 2 S is contractible. As a consequence, there exists a unique, up to contractible indeterminacy, morphism

◆ : ⇤ ! Sh ét (X)
in the 1-category Pro(S fc ). Precomposition along ◆ induces a functor of 1-categories

ev : Fun ECat1 Sh ét (X), Perf(Z) ! Fun ECat1 ⇤, Perf(Z) ' Perf(Z) 2 Cat 1 .
The important observation is illustrated by the following lemma:

Lemma 1.5.2.3. Let Z 2 Afd op k be a derived k-affinoid space which we suppose further to be n-truncated for a given integer n 1. Let M 2 Perf(Z) be a perfect module on Z and let. Then the fiber of the functor

ev : Fun ECat1 Sh ét (X), Perf(Z) ! Perf(Z)
at M 2 Perf(Z) is naturally equivalent to the mapping space

Map Mon E 1 (Top na ) ⌦Sh ét (X), End(M ) 2 S
where End(M ) denotes the ind-pro-endomorphism space of M .

Remark 1.5.2. Whenever Z is discrete and M ' (Z) n we recover the space of continuous representations

⇢ : ⇡ ét 1 (X) ! GL n A , A ' (Z).
Lemma 1.5.2.7. Let Z := Sp(A) 2 Afd k be an ordinary k-affinoid space. Then we have a natural equivalence of spaces

LocSys `,n (X)(A) ' RLocSys `,n (X)(A).

Notation 1.5.2.8. Following our convention, we will denote RLocSys `,n (X) by LocSys `,n (X).

As promised we state an explicit computation of the cotangent complex of LocSys `,n (X) Proposition 1.5.2.9. Let Z 2 dAfd k and let ⇢ 2 LocSys `,n (X)(Z) be a continuous `-adic representation. Then

LocSys `,n (X) admits an analytic cotangent complex at ⇢ and we have the following natural equivalence

L an LocSys `,n (X),⇢ ' C ⇤ ét X, Ad(⇢) _ [ 1]
in the derived 1-category Mod k .

Theorem 1.5.2.9 implies that deformations of ⇢ are classified by the étale homology complex C ⇤ ét X, Ad(⇢)

_ [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. Moreover, Porta-Yu Yue representability theorem, [PY17a, Theorem 7.1] implies the following: Theorem 1.5.2.10. The derived moduli stack LocSys `,n (X) : dAfd op k ! S is representable by a geometric stack with respect to the geometric context dAfd k , ⌧ ét , P sm , i.e. LocSys `,n (X) is representable by derived k-analytic stack.

Construction 1.5.2.11. Let ⇢ : ⇡ ét 1 (X) ! BGL n (F `r ) be a continuous representation. We can consider at the formal moduli problem Def ⇢ : CAlg sm F `r ! S given on objects by the formula

A 2 CAlg sm F `r 7 ! LocSys `,n (X)(A) ⇥ LocSys `,n (X)(F F `r ) {⇢} 2 S.
The functor Def ⇢ paramatrizes continuous deformations of ⇢ with values in small derived F `r -algebras. Such functor satisfies the conditions of Lurie-Schlessinger Theorem. Therefore it is pro-representable by a Noetherian derived W (F `r )-algebra, A ⇢ , augmented over F `r and complete with respect with the maximal ideal

m ⇢ := ker ⇡ 0 (A ⇢ ) ! F `r .
Let k := Frac(W (F `r )). By construction, we can consider Def ⇢ as a functor defined on the 1-category CAlg ad k . Moreover, such functor is representable by Spf(A ⇢ ), which is a locally admissible derived k -adic scheme. In this case, we can take its rigidification

Def rig ⇢ 2 dAn k ,
which is a derived k-analytic stack. We have a canonical morphism Def rig ⇢ ! LocSys `,n (X) which is roughly described by sending a continuous deformation of ⇢ to its corresponding continuous `-adic representation. Varying

⇢ : ⇡ ét 1 (X) ! GL n (F `r ) we obtain a canonical morphism a ⇢ Def rig ⇢ ! LocSys `,n (X) (1.5.2.3)
in the 1-category dSt dAfd, ⌧ ét , P sm of derived k-analytic stacks.

The derived structure on LocSys `,n (X) allow us to prove the following geometric result:

Proposition 1.5.2.12. The canonical morphism displayed in (1.5.2.3) is an étale admissible inclusion of subanalytic derived k-analytic stacks.

One could ask if the morphism (1.5.2.3) is an equivalence of geometric stacks. The following example illustrates that this is not is the case in general:

Example 1.5.2.13. Let G = Z p and A = Q p hT i the Tate algebra in one variable. Consider the continuous representation ⇢ : Z p ! Q p hT i determined by the association

1 2 Z p 7 !  1 T 0 1 .
Then ⇢ 2 LocSys p,2 (G)(Q p hT i) but does not belong to the disjoint union `⇢ Def rig ⇢ ✓ LocSys `,n (G).

However, as Theorem 1.5.2.12 suggests the derived k-analytic stack LocSys `,n (X) is highly disconnected. It would be desirable to have a way to glue together the formal neighborhoods Def rig ⇢ together. One could state it more precisely as a conjecture: Conjecture 1.5.2.14. There exists a (possibly ind-)derived k-analytic stack LocSys gl `,n (X) and a morphism of derived k-analytic stacks ⇡ : LocSys `,n (X) ! LocSys gl `,n (X) such that ⇡ is an equivalence at closed points and it induces an equivalence of cotangent complexes at closed points. Moreover, the moduli stack LocSys gl `,n (X) is equipped with an endomorphism, F , which is compatible with Frob ⇤ : LocSys `,n (X) ! LocSys `,n (X), i.e. we have a commutative diagram

LocSys `,n (X) LocSys `,n (X) LocSys gl `,n (X) LocSys gl `,n (X) Frob ⇤ ⇡ ⇡ F
in the 1-category dSt dAfd k , ⌧ ét , P sm . Moreover, LocSys gl `,n (X) is almost of finite presentation and in particular it admits finitely many connected components.

Open case

When X is assumed to be a smooth scheme over k = k of positive characteristic p 6 = `its étale fundamental group ⇡ ét 1 (X) is not topologically finitely generated (except if we assume X proper). In this case we have a short exact sequence of profinite groups

1 ! ⇡ w 1 (X) ! ⇡ ét 1 (X) ! ⇡ t 1 (X) ! 1
where ⇡ w 1 (X) denotes the wild fundamental group of X a pro-p profinite group and ⇡ t 1 (X) its tame fundamental group which is topologically finitely generated. The tame fundamental group parametrizes tamely ramified at infinity coverings of X. We cannot expect the full stack LocSys `,n (X) is representable as the profinite group ⇡ ét 1 (X) is too big. However, by bounding the ramification at infinity we can consider the substack

LocSys `,n, (X) ,! LocSys `,n (X)
which parametrizes continuous representations ⇢ : Sh ét (X) ! BGL n (A) such that the restriction ⇢ |⇡ w 1 (X) factors through a finite quotient q : ⇡ w 1 (X) ! . We have the following result Proposition 1.5.3.1. The moduli stack LocSys `,n, (X) is representable by a derived k-analytic stack.

Shifted symplectic structure

Let X be a smooth and proper scheme over a field k = k of positive characteristic p > 0. As X is proper, Poincaré duality for étale cohomology implies that we have a non-degenerate bilinear pairing

C ⇤ ét X, Q ` ⌦ C ⇤ ét X, Q ` _ ! Q `[ 2d],
where d = dim X. When X is non-proper we should replace étale cohomology with étale cohomology with support. Thanks to the projection formula, given a continuous representation

⇢ : ⇡ ét 1 (X) ! GL n (A), A 2 Afd k ,
one still obtains a non-degenerate pairing of the form

C ⇤ ét X, Ad(⇢) ⌦ C ⇤ ét X, Ad(⇢) _ ! Q `[ 2d], (1.5.4.1)
where Ad(⇢) := ⇢ ⌦ ⇢ _ denotes the adjoint representation associated to ⇢. It is possible to give a more conceptual construction of the pairing introduced above as follows: the 1-category Fun ECat1 Sh ét (X), Perf(A) is a rigid symmetric monoidal, that is every object is dualizable in Fun ECat1 Sh ét (X), Perf(A) . Therefore, given ⇢ as above one has a natural trace morphism

tr : Ad(⇢) := ⇢ ⌦ ⇢ _ ! 1,
where 1 denotes a unit for the symmetric monoidal structure on Fun ECat1 Sh ét (X), Perf(A) . As Ad(⇢) 2 Fun ECat1 Sh ét (X), Perf(A) is an E 1 -monoid object we have a canonical multiplication morphism We can identify the above composite with the canonical map displayed in (1.5.4.1). Define O : dAfd op k ! CAlg k as the sheaf on the étale site dAfd k , ⌧ ét given on objects by the formula

mult : Ad(⇢) ⌦ Ad(⇢) ! Ad(⇢). ( 1
Z 2 dAfd k 7 ! O(Z) := (Z) 2 CAlg k . The canonical map C ⇤ ét X, Q ` ! Q `[ 2d]
in the derived 1-category Mod Q `induces by the projection formula on `-adic cohomology a canonical morphism

C ⇤ ét X, A ! A[2d]
We obtain thus a non-degenerated pairing at the level of the cotangent complex of LocSys `,n (X) at ⇢

! ⇢ : L an LocSys `,n (X),⇢ ⌦ L an LocSys `,n (X),⇢ ! O Sp(A)
The results of [Toë18,[START_REF] Bhatt | The pro-étale topology for schemes[END_REF] imply that the pairing ! :

2 ^Lan LocSys `,n (X) ! O is closed, i.e. it
C ⇤ ét (X, Ad(⇢) [1] ⌦ C ⇤ ét X, Ad(⇢) ! (Z)[2 2d]
in `-adic cohomology.

Corollary 1.5.4.2. Let ⇢ 2 LocSys `,n (X)(Z), then the shifted symplectic form ! 2 HC LocSys `,n (X) induces an equivalence T an LocSys `,n (X) ' L an LocSys `,n (X) [2 2d] between the tangent and cotangent complexes on LocSys `,n (X).

Analytic HKR theorem

The results in this section were first study in a joint collaboration work between M. Porta and F. Petit. I thank both of them for letting me take part on the project.

Main results

Let k be a field of characteristic 0. In the setting of derived algebraic geometry the structured HKR theorem was first proved in [TV15]. More precisely, the HKR theorem states that there is an equivalence of 1-categories

S 1 -CAlg k ' CAlg k[✏]
where the left hand side denotes the 1-category of derived k-algebras equipped with an action of the circle S 1 2 S, whereas the right hand side denotes the 1-category of derived k[✏] := k k[1]-algebras. As a consequence one has the following global results:

Theorem 1.6.1.1 ( [TV15]
). Let X be a derived algebraic scheme over a field k of characteristic 0. Then one has an equivalence of derived algebraic stacks X ⇥ X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] where the left hand side denotes the derived fiber product of X with itself over X ⇥ X via the diagonal map and the right hand side denotes the 1-shifted tangent bundle on X. Moreover the above equivalence is compatible with the canonical projection to X.

It would be desirable to have an analytic analogue of the above result. In a joint work with M. Porta and F. Petit the authors prove: Theorem 1.6.1.2. Let k be the field C of complex numbers or a non-archimedean field of characteristic 0 with a non-trivial valuation. Let X be a k-analytic space. Then there is an equivalence of derived k-analytic spaces X ⇥ X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] compatible with the canonical projection to X. Suppose X is a derived k-affinoid space. Let A := (X, O X ). Then Theorem 1.6.1.2 implies that we have an equivalence of simplicial algebras

A b ⌦ A b ⌦ k A A ' Sym an A L an A [1]
where Sym an A denotes the analytification relative to A of the algebraic Sym A . Theorem 1.6.1.2 is a consequence of the following more general result: Theorem 1.6.1.3. There are 1-categories k[✏]-AnRing k of mixed analytic rings and S 1 -AnRing k of S 1 -equivariant analytic rings. These 1-categories are equivalent compatibly with their forgetful functors to AnRing k .

The 1-category S 1 -AnRing k is defined as

S 1 -AnRing k := Fun BS 1 , AnRing k .
By a formal argument, the 1-category S 1 -AnRing k is canonically monadic over AnRing k . Let us denote the associated monad by T S 1 . However, the construction of the 1-category k[✏]-AnRing k is more involved. We need thus to assume that there exists an 1-category k[✏]-AnRing k equipped with a functor

U ✏ : k[✏]-AnRing k ! AnRing k
such that U ✏ is conservative, commutes with sifted colimits and it admits a left adjoint

DR : AnRing k ! k[✏]-AnRing k
such that for every A 2 AnRing k there exists a canonical equivalence

U ✏ DR(A) ' Sym an A L an [1] .
In particular, U ✏ exhibits k[✏]-AnRing k as monadic over AnRing k . Let us denote the corresponding monad by

T ✏ .
The structured HKR theorem can be stated as:

Theorem 1.6.1.4 (Structured analytic HKR theorem). The monads T S 1 and T ✏ are equivalent as monads over AnRing k . In particular, there exists an equivalence of 1-categories

S 1 -AnRing k ' k[✏]-AnRing k
compatible with the forgetful functors to AnRing k .

Remark 1.6.1.5. Theorem 1.6.1.4 implies both Theorem 1.6.1.2 and Theorem 1.6.1.3.

Main results

For the reader's convenience we list the main results presented in the present thesis. The reader can find a more precise formulation of these in the body of the text.

Theorem 1.7.0.1. Let T ad (k ) denote the k -adic pregeometry and X an 1-topos. Then the exists a canonical functor fCAlg k (X) ! CAlg ad k (X) which is an equivalence when restricted to topologically almost of finite presentation local T ad (k )-structures on X.

dfSch[S 1 ] ' dAn 0 k in the 1-category Cat 1 .
Here dfSch ✓ dfDM k denotes the full subcategory spanned by admissible derived k -adic schemes, S the saturated class of generically strong morphisms and dAn 0 k ✓ dAn k the full subcategory spanned by quasi-paracompact and quasi-separated derived k-analytic spaces.

Theorem 1.7.0.5. Let X 2 dAn k be a quasi-compact and quasi-separated derived k-analytic space and X 2 dfDM k a formal model for X. Then we have an equivalence of stable 1-categories

Coh + (X) ' Coh +, (X)
where Coh +, (X) denotes the Verdier quotient of the diagram

Coh + nil (X) ! Coh + (X) ! Coh + (X)
computed in the 1-category Cat st 1 .

Theorem 1.7.0.6. Let X be a quasi-compact and quasi-separated derived k-analytic stack. Then the derived Hilbert stack RHilb(X) is representable by a derived k-analytic stack.

Theorem 1.7.0.7. Let X be a smooth over an algebraically closed field. Then there exists a moduli functor LocSys `,n (X) : dAfd op k ! S which is given on objects by the formula

Z 2 dAfd op k 7 ! Map cont Sh ét (X), BGL n ( (Z))
where Sh ét (X) 2 Pro(S fc ) denotes the étale homotopy type of X and Map cont denotes the space of morphisms

⇢ : Sh ét (X) ! BGL n ( (Z))
which preserve the canonical topologies on both Sh ét (X) and BGL n ( (Z)). Moreover, given a finite quotient ⇡ w 1 (X) ! of the wild fundamental group of X, there exists a derived stack

LocSys `,n, (X) 2 dSt dAfd k , ⌧ ét which parametrizes ⇢ : Sh ét (X) ! BGL n ( (Z)) whose restriction to ⇡ w 1 (X) factor through the quotient morphism ⇡ w 1 (X) ! .
Theorem 1.7.0.8. Let X be a smooth and proper scheme over an algebraically closed field. Then the derived stack LocSys `,n (X) 2 dSt(dAfd op k , ⌧ ét ) is representable by a derived geometric stack. Moreover, given ⇢ 2 LocSys `,n (X) its analytic cotangent complex at ⇢ is given by

L an LocSys `,n (X),⇢ ' C ⇤ ét X, Ad(⇢) _ [ 1] 2 Mod k .
(1.7.0.1)

In the case where X is a smooth non-proper scheme over an algebraically closed field of characteristic p > 0 (different than the residual characteristic of k), then the moduli

LocSys `,n, (X) 2 dSt dAfd k , ⌧ ét
is representable by a derived geometric stack. Moreover, the formula displayed in (1.7.0.1) holds for the analytic cotangent complex of LocSys `,n, (X).

Theorem 1.7.0.9. Let X be a proper and smooth scheme over an algebraically closed field. Then the moduli stack LocSys `,n (X) admits a natural shifted symplectic structure ! 2 A 2,cl (LocSys `,n (X)). Moreover, given ⇢ 2 LocSys `,n (X)(Z) the underlying 2-form on ⇢ coincides with the Poincaré duality morphism

! ⇢ : C ⇤ ét (X, Ad(⇢))[1] ⌦ C ⇤ ét (X, Ad(⇢))[1] ! (Z)[2 2d],
where d = dim(X).

Theorem 1.7.0.10. Let X be a derived k-analytic space. Then one has an equivalence of derived k-analytic stacks

X ⇥ X⇥X X ' TX[ 1],
compatible with the projection to X. In particular, if we assume further that X is a derived k-affinoid space and we let A := (X, O X ) we have an equivalence of derived k-algebras

A b ⌦ A b ⌦A A ' Sym an A (L an [1]).
Theorem 1.7.0.11. There are 1-categories k[✏]-AnRing k and S 1 -AnRing k of mixed derived k-analytic rings and S 1 -equivariant derived k-analytic rings, respectively. Moreover, these 1-categories are monadic and comonadic over AnRing k and there exists an equivalence of 1-categories

S 1 -AnRing k ' k[✏]-AnRing k which fits into a commutative diagram S 1 -AnRing k k[✏]-AnRing k AnRing k
of monads over AnRing k .

Notations and Conventions

We shall denote k a non-archimedean field equipped with a non-trivial valuation, k its ring of integers and sometimes we will use the letter t 2 k to denote a uniformizer for k. We denote An k the category of strict k-analytic spaces and Afd k the full subcategory spanned by strict k-affinoid spaces and we adopt the convention that whenever we mention k-affinoid or k-analytic space we mean strict k-affinoid and strict k-analytic space, respectively. We denote fSch k the category of quasi-separated formal schemes over the formal spectrum Spf(k ), where we consider k equipped with its canonical topology induced by the valuation on k. In order to make clear that we consider formal schemes over Spf(k ), we shall often employ the terminology k -adic scheme to refer to formal scheme over Spf k .

Let n 1, we shall make use of the following notations:

A n k := Spec k[T 1 , . . . , T m ], A n k := Spf(k hT 1 , . . . , T m i) and A n k := A n k an , B n k := Sp(khT 1 , . . . T m i)
, where ( ) an denotes the usual analytification functor ( ) an : Sch k ! An k , see [Ber93a]. We denote by GL an n the analytification of the usual general linear group scheme over k, which associates to every k-affinoid algebra A 2 Afd k the general linear group GL n (A) with A-coefficients.

In this thesis we extensively use the language of 1-categories. Most of the times, we reason model independently, however whenever needed we prove 1-categorical results using the theory of quasi-categories and we follows closely the notations in [Lur09b]. We use caligraphic letters C, D to denote 1-categories. We denote Cat 1 the 1-category of (small) 1-categories. We will denote by S the 1-category of spaces, S fc the 1-category of finite constructible space, see [Lur09a, §3.1]. Let C be an 1-category, we denote by Ind(C) and Pro(C) the corresponding 1-categories of ind-objects and pro-objects on C, respectively. When C = S fc , the 1-category

Pro(S fc

) is referred as the 1-category of profinite spaces. Let R be a derived commutative ring. We will denote by CAlg R the 1-category of derived k-algebras. The latter can be realized as the associated 1-category to the usual model category of simplicial R-algebras.

We shall denote by CAlg ad the 1-category of derived adic algebras, introduced in [Lur16, §8.1]. Whenever R admits a non-trivial adic topology, we denote CAlg ad R := CAlg ad R/ the 1-category of derived adic Ralgebras, i.e. derived R-algebras equipped with an adic topology compatible with the adic topology on R together with continuous morphisms between these.

Let R be a field. We shall denote by CAlg sm R the 1-category of small augmented derived R-algebras. When R = k we denote by AnRing sm k the 1-category of small augmented derived k-analytic rings over k, which is naturally equivalent to CAlg sm k , see [Por15a, §8.2]. Let R be a discrete ring. We denote by CAlg R the 1-category of ordinary commutative rings over R. When R admits an adic topology we shall denote CAlg ad,R ✓ CAlg ad R the full subcategory spanned by discrete derived adic R-algebras. Let R denote a derived ring. We denote Mod R the derived 1-category of R-modules and Coh + (X) ✓ Mod R the full subcategory spanned by those almost perfect R-modules. We need sometimes to enlarge the starting Grothendieck universe, and we often do not make explicit such it procedure. Fortunately, this is innocuous for us. We will usually employ caligraphic letters X, Y, Z to denote 1-topoi. The 1-category of 1-topoi together with geometric morphisms between these is denoted Top R . Caligraphic letters such as O, A, B are often employed to denote structures on an 1-topos. We will denote by T ad (k ) and T an (k) the adic and analytic pregeometries, respectively. Let X 2 Top R be an 1-topos, we denote by fCAlg k (X) := Str loc T ad (k ) (X) and AnRing k (X) := Str loc Tan(k) (X). We will denote by dA↵ k , ⌧ ét , P sm the algebraic geometric context and we denote by dSt dA↵ k , ⌧ ét , P sm the 1-category of derived geometric stacks with respect to dA↵ k , ⌧ ét , P sm . Similary, whenever k denotes either the field C of complex numbers or a non-archimedean field we will denote by dA↵ k , ⌧ ét , P sm the analytic geometric context and correspondingly dSt dAn k , ⌧ ét , P sm the 1-category of derived geometric stacks with respect to the analytic geometric context.

Chapter 2

Brief overview of derived k-analytic geometry

For the purposes of clarity we introduce a small section reviewing the main foundational results in derived kanalytic geometry, proved in [PY16a,PY17a].

Derived rigid analytic geometry

Let k be a non-archimedean field with a non-trivial valuation. In [PY16a,PY17a] M. Porta and T. Yu Yue introduced the foundations of derived k-analytic geometry. Roughly speaking, a derived k-analytic space consists of a couple (X, O alg

) where X is an 1-topos and O alg is a locally ringed sheaf on X having an additional k-analytic structure.

Definition 2.1.0.1. Let T an (k) denote the full subcategory of An k spanned by smooth k-analytic spaces. We endow T an (k) with étale Grothendieck topology. We define a class of admissible morphisms on T an (k) as the class of étale morphisms on T an (k). Definition 2.1.0.2. Let X be an 1-topos. We say that a functor O : T an (k) ! X is a local T an (k)-structure on X if the following conditions are satisfied:

(i) The functor O : T an (k) ! X commutes with finite produces in T an (k);

(ii) The functor O : T an (k) ! X commutes with pullbacks along admissible morphisms, i.e. given a commutative diagram

U V U 0 V 0 g (2.1.0.1) in the category T an (k) such that g : V ! V 0 is an admissible morphism then the induced commutative diagram O(U ) O(V ) O(U 0 ) O(V 0 ) g in X is a pullback diagram. (iii) Let ì U i ! U be an étale covering in T an (k), then the corresponding morphism ì O(U i ) ! O(U ) is an
effective epimorphism in the 1-topos X. We say that a morphism

↵ : O ! O 0 between local T an (k)-structures on X is local if the for admissible morphism g : V ! U in T an (k) the induced commutative diagram O(V ) O 0 (V ) O(U ) O 0 (U ) is a pullback diagram in X.
Notation 2.1.0.3. Let X be an 1-topos. The subcategory of Fun T an (k), X spanned by local T an (k)-structures and local morphisms between these is denoted by AnRing k (X).

Remark 2.1.0.4. Let O : T an (k) ! X denote a local T an (k)-structure on X. Condition (i) in Theorem 2.1.0.2 implies that evaluation on the affine line induces a sheaf O(A 1 k ) 2 X which can be promoted to a CAlg k (X)-valued sheaf on X. Similarly, the evaluation on the closed unit disk induces a CAlg k -valued sheaf O(B 1 k ) on X.

Definition 2.1.0.5. Let O : T an (k) ! X be a local T an (k)-structure on X. We define its underlying algebra as the

CAlg k -valued sheaf O alg := O(A 1 k ) on X.
This association is functorial and the corresponding functor is denoted

( ) alg : AnRing k (X) ! CAlg k (X)
and referred to as the underlying algebra functor. Example 2.1.0.8. Let X 2 An k denote an ordinary k-analytic space. To X we can associate X X := Shv(X ét ) ^, the hypercompletion of the 1-topos of sheaves on the (quasi-)étale site of X. We can define a T an (k)-structure on X as follows: given V 2 T an (k) we associate it the sheaf O(V ), on X, defined on objects via the formula

U 2 X ét 7 ! Map An k (U, V ) 2 S.
Notice that when V = A 1 k , denotes the k-analytic affine line, the sheaf O(A 1 k ) coincides with the usual sheaf of sections on X. Definition 2.1.0.9. A derived k-analytic space consists of a couple X = (X, O) where X is an 1-topos and O : T an (k) ! X is a T an (k)-structure on X satisfying the following conditions:

(i) The 0-truncation t 0 X := X, ⇡ 0 (O) is equivalent to an ordinary k-analytic space via Theorem 2.1.0.8. (ii) For each i > 0, the homotopy sheaf ⇡ i O alg is a coherent sheaf over X, ⇡ 0 (O) . Notation 2.1.0.10. We will denote by dAn k the 1-category of derived k-analytic spaces.

The theory of derived k-analytic geometry is robust in the sense that in practice the main results of derived algebraic geometry do admit analogues in the k-analytic setting. We cite some of the most relevant results in derived k-analytic geometry: Theorem 2.1.0.11 (Gluing along closed immersions, Theorem 6.5 [PY17a]). Consider the following pushout diagram

X X 0 Y Y 0 i j q
in the 1-category Top R (T an (k)). Suppose further that i and j are closed immersions and X, X 0 , Y are derived k-analytic spaces. Then Y 0 is itself a derived k-analytic space.

Theorem 2.1.0.12 (Existence of an analytic cotangent complex). Let X := (X, O X ) 2 dAn k be a derived kanalytic space and suppose we are given a morphism f : X ! Y . Consider the relative analytic derivations functor Der an X/Y ( ) : Mod O X ! S given on objects by the formula

M 2 Mod O X 7 ! Der an X (M ) := Map AnRing k (X) f 1 O Y //O X (O X , O X M ) 2 S,
where O X M denotes the trivial square zero extension of O X by M , see [PY17a,[START_REF] Cadoret | The fundamental theorem of Weil II for curves with ultraproduct coefficients[END_REF]. Then Der an X/Y is corepresentable. More precisely, there exists an object L an X/Y , which we refer to the relative analytic cotangent complex of f : X ! Y such that for every M 2 Mod O X there exists a natural equivlence

Map AnRing k (X) f 1 O Y //O X (O X , O X M ) ' Map Mod O X ⇣ L an X/Y , M ⌘
in the 1-category of spaces S. Whenever f = Id X we refer to L an X := L an X/Y as the absolute analytic cotangent complex of X.

Moreover, the analytic cotangent complex satisfies:

(i) Let X 2 DM k denote a derived Deligne-Mumford stack over k. Then one has a natural equivalence

L X an ' L an X in the 1-category Coh + (X)
, where L X denotes the algebraic cotangent complex, introduced in [Lur12c, §7.3.5].

(ii) Let f : X ! Y and g : Y ! Z be morphisms between derived k-analytic spaces. Then there exists a fiber sequence of relative cotangent complexes of the form

f ⇤ L an Y /Z ! L an X/Z ! L an X/Y
in the 1-category Coh + (X).

(iii) Suppose we have a pullback square in the 1-category dAn k

X 0 Y 0 X Y. g f
Then one has a natural equivalence

g ⇤ L an X 0 /Y 0 ' L an X/Y in the 1-category Coh + (X 0 ).
Theorem 2.1.0.13 (Compatibility with Postnikov towers, Corollary 5.44 [PY17a]). Let X := (X, O X ) 2 dAn k be a derived k-analytic space. Then for every n 0, the canonical map t n X ,! t n+1 X is an analytic square zero extension. More precisely, we have a pushout diagram

t n X[⇡ n+1 O X [n + 2]] t n X t n X t n+1 X d0 d
in the 1-category dAn k , where

t n X[⇡ n+1 O X [n + 2]] := X, O X ⇡ n+1 (O X )[n + 2] 2 dAn k denotes the trivial square extension of X by ⇡ n+1 (O X )[n + 2] 2 Mod O X .
Moreover, d 0 and d denote the trivial square-zero extension and a suitable analytic derivation

d : L an t n X ! ⇡ n+1 (O X )[n + 2]
in the 1-category Mod O X , respectively.

Theorem 2.1.0.14 (Representability theorem, Theorem 7.1 [PY17a]). Let F 2 dSt dAfd k , ⌧ ét denote a derived stack. Then the following assertions are equivalent:

(i) F is a geometric n-stack with respect to the geometric context dAfd k , ⌧ ét , P sm .

(ii) F is compatible with Postnikov towers, has a global cotangent complex and its truncation t 0 F is representable by an n-geometric stack with respect to the geometric context Afd k , ⌧ ét , P sm .

We refer the reader to [PY16c,[START_REF] António | p-adic derived formal geometry and derived Raynaud localization Theorem[END_REF] for the notions of geometric context and geometric stack with respect to a given geometric context.

The above results were proved by M. Porta and T. Yu Yue. They constitute an extensive review of derived methods in the context of k-analytic geometry. However, certain results of classical k-analytic geometry still did not have derived analogues prior to the current thesis. They constitute mainly the existence of formal models for k-analytic spaces and its applications in k-analytic geometry. Certain of these lacking results were desired in order to apply the techniques of derived k-analytic geometry to the study of certain problems coming from representation theory. We shall exemplify one such application, which is an ubiquitous theme in the current thesis.

Example 2.1.0.15. Let X be a smooth scheme over an algebraically closed field of positive characteristic p > 0. Let `6 = p be a prime number and suppose we are given a continuous representation

⇢ : ⇡ ét 1 (X) ! GL n Q ` .
As ⇢ is continuous we can suppose that ⇢ factors through a finite extension k/Q `. One would like to understand the space of continuous deformations of ⇢. These should correspond to continuous group representations e ⇢ : ⇡ ét 1 (X) ! GL n (A) where A 2 AnRing sm k , as we are concerned with continuous deformations of ⇢. In such case, one would like to consider the formal moduli problem F : AnRing sm k ! S given on objects by the formula

A ! k 2 AnRing sm k 7 ! Map cont B⇡ ét 1 (X), BGL n (A) ⇥ Map cont (B⇡ ét 1 (X),BGLn(k)) {⇢} 2 S. (2.1.0.2)
Unfortunately, we still do not have a precise definition of the right hand side of (2.1.0.2). We would like to define it as the space of continuous group-like homomorphisms

e ⇢ : ⇡ ét 1 (X) ! Aut(A n ) (2.1.0.3)
such that its restriction along the morphism A ! k in the 1-category AnRing sm k coincides with ⇢, up to equivalence. However, we do not know what continuity means in this context. Indeed, A 2 AnRing sm k corresponds to a functor A : T an (k) ! S satisfying certain admissibility conditions captured in Theorem 2.1.0.2. Such k-analytic structure on A do not produce any sort of topological data. Therefore, we need to interpret A differently in order to be able to define continuous morphism of group-like objects as in (

⇢ : ⇡ ét 1 (X) ! GL n (A).
Moreover, when A = k a continuous representation ⇢ : ⇡ ét 1 (X) ! GL n (k) corresponds to the usual notion of `-adic continuous representations of ⇡ ét 1 (X).

39

Contents 3.1 Introduction

Background material

Let k be a non-archimedean field of discrete valuation, k its ring of integers and let t 2 k be a fixed uniformizer for k. Denote fSch k the category of admissible k -adic formal schemes and An the category of k-analytic spaces.

There exists a rigidification functor ( ) rig : fSch k ! An such that every quasi-paracompact and quasi-separated k-analytic space X admits a formal model over Spf k . That is to say, there exists X 2 fSch k such that X rig ' X.

For this reason, one is able to understand the analytic structure on X through a formal model X for X. The following is a classical result proved by Raynaud:

Theorem 1 (Raynaud, Theorem 8.4.3 [Bos05] ). The functor ( ) rig : fSch k ! An is a localization functor. More specifically, the functor ( ) rig : fSch k ! An factors through the localization of fSch k at the class of admissible blow ups, S. Moreover such functor induces an equivalence of categories

fSch k [S 1
] ! An 0

where An 0 ✓ An denotes the full subcategory of quasi-paracompact quasi-separated k-analytic spaces.

?? 1 it allows to use methods from algebraic geometry in order to establish certain results in the context of rigid analytic geometry. For instance, ?? 1 is useful to study flatness conditions for k-analytic spaces and base change theorems in the setting of k-analytic geometry. Raynaud's theory allows to bypass this problem the intrinsic analytic difficulties by reducing this problem to its analogue at the formal level. The latter situation can then be dealt using techniques from algebraic geometry.

Main results

The same situation occurs in the context of derived k-analytic geometry. Derived k-analytic geometry was developed by M. Porta and T. Yu Yue in [PY16a,PY17a]. In [Lur16,[START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF] the author introduces and studies at length derived and spectral formal geometry. Our main goal in this text is to prove an analogue of ?? 1 in the derived setting. However, in order to state a derived analogue of ?? 1 one needs another crucial ingredient, namely the existence of a derived rigidification functor. Inspired by the construction of the derived analytification functor [PY17a, §3], we will provide a construction of a derived rigidification functor. In order to so, we need to develop a structured spaces approach to derived formal geometry over Spf k . This is done in §2: we develop a theory of derived t-adic formal geometry by considering certain T ad (k )structured spaces. Therefore, we will consider couples (X, O) where X is an 1-topos and O : T ad (k ) ! X is a local T ad (k )-structure. To such a pair we can functorially associate a locally ringed 1-topos (X, O alg

). However, this construction loses information.

In general, the T ad (k )-structure O : T ad (k ) ! X encodes more information than its algebraic counterpart O alg 2 CAlg k (X). For example, one can show that ( ) alg factors through the canonical functor CAlg ad k (X) ! CAlg k (X), where CAlg ad k (X) denotes the 1-category of k -adic algebra objects on the 1-topos X.

More specifically, CAlg ad k (X) corresponds to the 1-category whose objects are objects O 2 CAlg k (X) together with an adic topology on ⇡ 0 (O) compatible with the adic topology on k and continuous morphisms between these. Fortunately, we are able to fully understand the difference between the 1-categories fCAlg k (X) and CAlg k (X). Theorem 3.1.2.1 (Theorem 3.3.2.4). Let X be an 1-topos and consider the underlying algebra functor

( ) alg : fCAlg k (X) ! CAlg k (X),
given on objects by the formula

(X, O) 7 ! (X, O alg
). Then, this functor factors through the functor CAlg ad k (X) ! CAlg(X) and the induced functor

( ) ad : fCAlg k (X) ! CAlg ad k (X)
. is fully faithful and moreover an equivalence of 1-categories when restricted to those strictly Henselian objects topologically almost of finite presentation. Theorem 3.1.2.1 implies that the 1-category of T ad (k )-structured spaces (X, O) whose O alg 2 CAlg k (X) is topologically almost of finite presentation can be recovered as locally ringed 1-topoi (X, O alg

) such that ⇡ 0 O alg comes equipped with an adic topology compatible with the t-adic topology on k . This can be regarded as a rectification type result for T ad (k )-structured spaces.

We will give a definition of derived formal k -adic Deligne-Mumford stacks over k in terms of T ad (k )structured spaces and show that this notion agrees with the notion introduced in [Lur16, §8]. We then proceed to study k -adic Postnikov tower decompositions and the k -adic cotangent complex with respect to maps between derived k -adic Deligne-Mumford stacks, which, to the author's best knowledge, has never been addressed before in the literature.

In §3 we define a rigidification functor

( ) rig : R Top (T ad (k )) ! R Top (T an (k))
which restricts to a functor ( ) rig : dfDM k ! dAn, where dfDM k denotes the 1-category of formal derived Deligne-Mumford stacks and dAn the 1-category of derived k-anlytic spaces. We prove that the derived rigidification functor ( ) rig coincides with the usual rigidification functor when restricted to the category of ordinary formal schemes.

Whenever Z 2 dAn is such that its 0-th truncation t 0 (Z) is an (ordinary) quasi-separated and quasi-paracompact k-analytic space, we prove: Theorem 3.1.2.2 (Theorem 6.2.3.15). Let Z 2 dAn be a quasi-paracompact and quasi-separated derived kanalytic space. There exists Z 2 dfDM such that one has an equivalence (Z) rig ' Z in the 1-category dAn, in other words Z admits a formal model Z 2 dfDM k .

Let dfSch k denote the full subcategory of dfDM k spanned by those X 2 dfDM k such that t 0 (X) is equivalent to an ordinary admissible quasi-paracompact formal scheme over k . We say that a morphism f : X ! Y in dfDM k is generically strong if for each i > 0, the induced map

⇡ i f 1 O Y ! ⇡ i O X is an equivalence in Coh + (X)
. Denote moreover dAn 0 ✓ dAn the full subcategory spanned by those X 2 dAn such that its 0-th truncation t 0 X is equivalent to a quasi-paracompact and quasi-separated ordinary k-analytic space. The following is a direct generalization of Raynaud's localization theorem in the derived setting: Theorem 2 (Theorem 3.4.4.10). Let S denote the saturated class generated by those morphisms f : X ! Y in dfSch k such that t 0 (f ) is an admissible blow up and generically strong. Then the rigidification functor

( ) rig : dfSch k ! dAn 0 k . factors through the localization 1-category dfSch k [S 1
] and the induced functor

dfSch k [S 1 ] ! dAn 0 k . is an equivalence of 1-categories.
Let us briefly sketch the proof of Theorem 3.4.4.10. In order to prove the statement it suffices to prove that given X 2 dAn as in Theorem 3.4.4.10 the comma category C X := dfSch X/ is contractible. We will prove a slightly stronger result, namely C X is a filtered 1-category. In order to illustrate the main ideas behind the proof it suffices to deal with lifting a morphism f : X ! Y in dAn to a morphism f + : X ! Y in dfSch such that (f + ) rig ' f as morphims in dAn.

The lifting is done by induction on the Postnikov tower of X. Suppose that X ' t 0 (X) in the 1-category dAn. Notice that ?? 1 implies that we can lift t 0 (f ) to a morphism f + 0 : X 0 ! Y 0 in the category fSch k . As X ! Y factors through the canonical morphism t 0 Y ! Y in dAn, we conclude by Theorem 3.1.2.2 together with ?? 1 that we can find a formal model for f : X ! Y , up to an admissible blow up at the level of 0-th truncations.

Let n 0 be an integer. Assume moreover that we are giving a morphism

(f + n ) : X n ! Y n in dfSch such that (f + n ) rig ' t n f : t n X ! t n Y .
Consider the (n + 1)-st step of the Postnikov tower, namely the pushout diagram

t n X[⇡ n+1 (O X )[n + 2]] t n X t n X t n+1 X
in the 1-category dAn. In order to proceed, we will need to know that the adic cotangent complex is compatible with the analytic one via rigidification. Namely, we have the following proposition:

Proposition 3.1.2.3. Let X 2 dfSch and denote X := X rig 2 dAn. Then the rigidification functor induces a canonical equivalence

L ad X rig ' L an X in the 1-category Coh + (X).
The induction hypothesis, together with the universal property of both the adic and analytic cotangent complexes plus refined results on the existence of formal models for almost perfect modules on X, proved in Appendix A, imply that we can extend the morphism f + n :

X n ! Y n to a diagram f + n f + n [⇡ n+1 (f ) + [n + 2]] ! t n f + n (3.1.2.1)
considered as an object in Fun ⇤ 2 0 , dfSch

1
, where ⇡ n+1 (f ) + 2 Coh + (X 0 ) 1 in (3.1.2.1) denotes a formal model for ⇡ n+1 (f ). By taking pushouts along ⇤ 2 0 we obtain the desired lifting f + n+1 : X n+1 ! Y n+1 of t n+1 (f ). The main technical difficulty of the proof comes from lifting higher coherences on diagrams of analytic derivations to suitable higher coherences of suitable diagrams of adic derivations. This is needed in order to extend (3.1.2.1) above in the case of more complex diagrams.

Related works

Let us give some examples of applications: it was proven in [Ant17a] that the moduli stack of continuous t-adic representations of a profinite group (topologically of finite generation) is representable by a geometric k-analytic stack. This object can be upgraded as a geometric derived k-analytic stack. This additional structure is crucial if one wants to obtain the correct cotangent complex and thus have a control of its obstruction theory. This additional structure led us to have a better understanding of the underlying geometry of such geometric derived k-analytic stack, in particular one is then able that it admits a shifted symplectic form.

However, the proof of the representability of such a derived k-analytic stack is not possible using only the techniques available from the structured spaces approach to derived k-analytic geometry, as in [PY16a]. The main drawback is that derived k-analytic spaces are defined as couples (X, O), where X denotes an (hypercomplete) 1-topos and O : T an (k) ! X consists of a T an (k)-local structure. This data should be interpreted as the given of a locally ringed space together with an additional structure, such structure consisting of the data of convergent t-adic holomorphic (derived) sections of the structure sheaf.

However such information does not provide directly any sort of topological structure on O, in contrast with the classical setting in which O corresponds to a sheaf of Banach k-algebras. Since [Ant17a] studies continuous representations of a profinite group and, more generally, of pro-homotopy types, one needs to be able to recover back this topological data at the derived level. ?? 2 provides us with an answer to this matter and it plays a crucial role in the proof of representability of [Ant17a].

So far, Raynaud's viewpoint in the derived setting already encountered other applications: in a joint work with F. Petit and M. Porta one proves an HKR Theorem in the context of derived k-analytic geometry and the theory of formal models proves to be useful in the proof of such statement. Another such application is a joint work with M. Porta: we show the representability of the derived Hilbert stack as a derived k-analytic stack in which the theory of formal models plays a crucial role.

Notations and conventions

Throughout the text, unless otherwise stated, k denotes a non-archidemean field of discrete valuation and k = {x 2 k : |x|  1} its ring of integers in k. We let t be a fixed uniformizer of k. Given an integer n 1, we will denote by k n the reduction modulo (t n ) of k . We denote fSch k the (classical) category of formal schemes (topologically) of finite presentation over k .

Let n 0 be an integer, we define k hT 1 , . . . , T n i as the sub-algebra of k [[T 1 , . . . , T n ]] consisting of those formal power series which f = ⌃ I a i T b I I , such that the coefficients a I ! 0 in k . Denote by A n k := Spf k hT 1 , . . . , T n i, B n k := SphT 1 , . . . , T n i the closed unit disk and A n k := A n k an the k-analytic affine n-space. We say that a morphism between two t-complete k -algebras A ! B is formally étale if, for each n 0, its mod t n reduction is an étale homomorphism of k /t n -algebras. We denote S the 1-category of spaces and R Top the 1-category of 1-topoi together with geometric morphisms between these. Let R be a commutative simplicial ring, we denote CAlg R its 1-category of derived R-algebras. Given an object B 2 CAlg R we denote by ⇡ i (B) the i-th homotopy group of the underlying space associated to B. We will denote Mod R the derived 1-category of R-modules, it can be considered as an 1-categorical upgrade of the usual (triangulated) derived category D(R) of R-complexes. Throughout the text we will employ homological convention, thus given M 2 Mod R we denote by ⇡ i (M ) := H i (M ) its i-th homology group. Given an 1-topos X we will denote CAlg R (X) := Str loc

T disc (R) (X), CAlg sh R (X) := Str loc Tét(R) (X), fCAlg k (X) := Str loc T ad (k ) (X)
and AnRing k (X) := Str loc Tan(k) (X). We will often denote a general pregeometry by the letter T. Moreover, whenever we refer to an object

(X, O) 2 Top R (T) we assume that O 2 Str loc T (X) is a local T-structure on X.

Acknowledgments

The author is extremely thankful to Mauro Porta for sharing many ideas, remarks and advices about the contents of the paper. The author is also very thankful to his advisor B. Toën and M. Robalo for valuable discussions.

3.2 Review on derived algebraic and analytic geometry

0 (f ) : ⇡ 0 (A) ! ⇡ 0 (B)
is an étale morphism of ordinary commutative rings and for each i > 0 the induced morphism

⇡ i (f ) : ⇡ i (A) ⌦ ⇡0(A) ⇡ 0 (B) ! ⇡ i (B)
is an isomorphism of ⇡ 0 (B)-modules. (i) Its 0-th truncation t 0 (X, O) := (X, ⇡ 0 (O)) is isomorphic to an ordinary scheme over k;

(ii) For every i 0, the higher homotopy sheaf ⇡ i (O) is a quasi-coherent sheaf on t 0 (X, O).

Remark 3.2.1.9. One can think of a pair (X, O) 2 dSch k as an infinitesimal deformation of the ordinary scheme (X, ⇡ 0 (O)) and the higher homotopy sheaves ⇡ i (O) encode the higher infinitesimal information.

Remark 3.2.1.10. One can realize the 1-category as a full subcategory of dLocRing k . For this reason, we have a canonical functor dLocRing k ! Fun A↵, S which associates to every (X, O) 2 dLocRing k the functor

⇣ Spec A 7 ! Map dLocRing k (Spec A, (X, O)) ⌘ 2 Fun A↵, S .
This provides a fully faithful embedding of the 1-category of derived schemes in the 1-category dSt dA↵, ⌧ ét .

Example 3.2.1.11. Let X be a usual scheme and Y , Z two full subschemes of X then we can define the derived intersection Y \ Z := Y ⇥ R X Z (in the ambient space X) as the derived scheme whose underlying topological space corresponds to the underlying topological space of the ordinary pullback, Y ⇥ X Z. Plus, the structure sheaf on Y \ Z coincides with the derived tensor product

O Y \Z := O Y ⌦ O X O Z .
It turns out that the 0-th truncation of Y \ Z coincides with the ordinary fiber product of Y with Z over X. More specifically, one has

⇡ 0 (O Y \Z ) ' Tor 0 O X (O Y , O Z ) and isomorphisms of coherent sheaves on Y ⇥ X Z, ⇡ i (O Y \Z ) ' Tor i O X (O Y , O Z ).
The Serre intersection formula implies that the Euler characteristic of the derived intersection Y \ Z agrees with the usual intersection number associated to the intersection of Y and Z inside of X.

Structured spaces approach

In [Lur11c], J. Lurie introduced the notion of a (spectral) scheme, and more generally (spectral) Deligne-Mumford stack via a structured spaces approach. Whenever k is a field of characteristic zero both approaches the functor of points and the structured spaces to derived algebraic geometry are equivalent. We review some of these notions which will be useful for our exposition. The reader is referred to [Lur11c] and [PY16a] for more details. Definition 3.2.2.1. We refer (X, O) to a couple as a ringed 1-topos whenever X is an 1-topos and O 2 CAlg(X) is a CAlg-valued sheaf on X. We say that a ringed 1-topos is a locally ringed 1-topos if for each geometric point x ⇤ : X ! S the CAlg-valued sheaf x 1 O on S can be identified with a local derived k-ring. Remark 3.2.2.2. Suppose given X a topological space. We can then form its associated 1-topos X := Shv(X) of S-valued sheaves on X. The locally ringed pair (X, O) induces naturally a locally ringed 1-topos (X, O), as O can be by construction promoted to a local CAlg-valued sheaf on X.

We now reformulate the notion of locally ringed 1-topos in terms of pregeometries: Definition 3.2.2.3. A pregeometry consists of an 1-category T equipped with a class of admissible morphisms and a Grothendieck topology, which is generated by admissible morphisms satisfying the following conditions:

(i) T admits finite products;

(ii) Pullbacks along admissible morphisms exist and are again admissible;

(iii) If f and g are morphisms in T such that g and g f are admissible then so is f . (iv) Retracts of admissible morphisms are again admissible.

We give a list of well known examples of pregeometries which will be useful later on.

Example 3.2.2.4.

(i) Let T disc (k) denote the pregeometry whose underlying category consists of affine spaces A n k and morphisms between these. The family of admissible morphisms is the family of isomorphisms in T disc (k) and we equip it with the discrete Grothendieck topology.

(ii) Let T Zar (k) denote the pregeometry whose underlying category has objects those affine schemes which admit an open embedding in some n-th affine space, A n k , whose admissible morphisms correspond to open immersions of schemes and the Grothendieck topology consists of usual Zariski topology.

(iii) Let T ét be the pregeometry whose underlying category is the full subcategory of the category affine schemes spanned by affine schemes étale over A n k , for some n. A morphism in T ét is admissible if and only if it is an étale morphism of affine schemes. Definition 3.2.2.5. Let T be a pregeometry and X an 1-topos. A T-local structure on X is defined as a functor between 1-categories O : T ! X satisfying the following conditions:

(i) The functor O preserves finite products;

(ii) For a pullback square of the form

U 0 X 0 U X f
in T where f is admissible then the square

O(U 0 ) O(X 0 ) O(f ) O(U ) O(X)
is a pullback square in X.

(iii) Given a covering {U ↵ ! U } in T consisting of admissible morphisms then the induced map

a O(U ↵ ) ! O(U ),
is an effective epimorphism in X.

A morphism O ! O 0 between T-local structures is said to be local if it is a natural transformation satisfying the additional condition that for every admissible morphism U ! X in T, the resulting diagram

O(U ) O 0 (U ) O(X) O 0 (X),
is a pullback square in X. We denote Str loc T (X) the 1-category of local T-structures on X together with local morphisms between these. T disc (X) ' Shv CAlg (X), where the latter denotes the 1-category of CAlg-valued sheaves on X. More explicitly, given a ringed 1-topos (X, O), we can promoted it naturally to a T disc -structured via the construction:

A n k 2 T disc 7 ! (O n 2 Shv (X) ' X) ,
where we forget the additional ringed structure on O.

(ii) Let O : T Zar (k) ! X be a T Zar (k)-local structure on the 1-topos. We can restrict it to a T disc (k)-structure on X via the natural inclusion functor T disc (k) ! T Zar which thus induces a CAlg-valued sheaf on

(i) The 0-truncation t 0 (X, O) := X, ⇡ 0 (O alg
) is equivalent to an (ordinary) Deligne-Mumford stack;

(ii) For each i > 0, the higher homotopy sheaf ⇡ i O alg is a quasi-coherent sheaf on (X, O).

Derived k-analytic geometry

Let k denote a non-archimedean field of non-trivial valuation. Derived k-analytic geometry as introduced in [PY16a] is a vast generalization of the classical theory of rigid analytic geometry. It is far more complicated to introduce derived k-analytic geometry through a functor of points approach. The main drawback comes from the fact that there is no reasonable description of the 1-category of (derived) affinoid spaces. For this reason, we prefer to adopt a structured spaces approach as in [PY16a]. We will review the basic definitions and we shall refer the reader to [PY16a,PY17a] for a detailed account of the foundational aspects of the theory.

Definition 3.2.3.1. Let T an (k) denote the pregeometry whose underlying category consists of those k-analytic spaces which are smooth and whose admissible morphisms correspond to étale maps between these. We equip T an (k) with the étale topology.

Construction 3.2.3.2. Let X be an ordinary k-analytc space and denote X ét the associated small étale site on X.

Let X := Shv ét (Xét) ^denote the hypercompletion of the 1-topos of étale sheaves on X. We can attach to X a T an (k)-structure on X as follows: given U 2 T an (k), we define the sheaf O(U ) 2 X by

X ét 3 V 7 ! Hom An (V, U ) 2 S.
As in the algebraic case, we can canonically identify O(A 1 k ) with the usual sheaf of analytic functions on X.

Definition 3.2.3.3. We say that T an (k)-structured 1-topos (X, O) is a derived k-analytic space if the following conditions are satisfied:

(i) X is hypercomplete and there exists an effective epimorphism `i U i ! 1 X on X verifying: (iii) For each index i and j 1, the ⇡ j (O alg |U i ) is a coherent sheaf over ⇡ 0 (O alg |U i )-modules on X i .

(ii) For each i, the couple (X |Ui , ⇡ 0 (O alg |U i )) is equivalent in the 1-category Top

Derived k -adic geometry

In this section we will introduce the k -adic pregeometry, T ad (k ), and study the corresponding theory of T ad (k )structured spaces. Let Top R (T ad (k )) denote the 1-category of T ad (k )-structured spaces. Our first goal is to make precise the assertion that a T ad (k )-structured 1-topos (X, O) can be realized as a locally ringed 1-topos (X, O alg

) together with an adic topology on ⇡ 0 O alg . We will prove such assertion in §2.1 in the case where O is almost of finite presentation. We will also extend the Spf-construction introduced in [Lur16, §8.2] to the context of T ad (k )-structured spaces. We will then proceed to a formal study of the theory of modules and obstruction theory in this context.

We show that Postnikov towers for T ad (k )-structured spaces exist and they are controlled by the adic cotangent complex. T ad (k ) (X) the 1-category of local T ad (k )-structures on X. Notation 3.3.1.3. We have canonical transformation of pregeometries, denoted

( ) t : T disc (k ) ! T ad (k ), ( ) t : T ét (k ) ! T ad (k )
obtained by performing completion along the t-locus. Precomposition along these transformations induce functors at the level of the 1-categories of structured 1-topoi:

( ) alg : Top R (T ad (k )) ! Top R (T disc (k )), ( ) sh : Top R (T ad (k )) ! Top R (T ét (k ))
which are determined by the association

(X, O) 2 Top R (T ad (k )) 7 ! (X, O alg ) 2 Top R (T disc (k )) (X, O) 2 Top R (T ad (k )) 7 ! (X, O sh ) 2 Top R (T ét (k )).
Let X 2 Top R be an 1-topos. Both functors ( ) alg : Top

R (T ad (k )) ! Top R (T disc (k )) and ( ) sh : Top R (T ad (k )) ! Top R (T ét (k ))
induce well defined functors at the level of 1-categories of structures on X:

( ) alg : fCAlg k (X) ! CAlg k (X),
( ) sh : fCAlg k (X) ! CAlg sh k (X) which we refer to as the underlying algebra functor and the underlying T ét (k )-structure functor, respectively. : fCAlg k (X) ! CAlg k (X) can be upgraded as a functor ( ) ad : fCAlg k (X) ! CAlg ad k (X) as follows: for each integer n 1 and for each A 2 fCAlg k (X), consider the canonical morphism

A alg ! A alg ⌦ k k n 2 CAlg k (X). Denote by I n := ker ⇡ 0 (A alg ) ! ⇡ 0 (A alg ⌦ k k n ) .
The sequence of ideals {I n } n 1 defines an adic structure on A alg which is moreover compatible with the t-adic topology on k . Moreover, for every morphism f : A ! B in fCAlg k (X) the forgetful f alg : A alg ! B alg is compatible with the adic topologies on both A alg and B alg : this can be checked at the level of ⇡ 0 in which case follows from the fact that every morphism A alg ! B alg ! B alg ⌦ k k n induces a unique, up to contractible space of choices, morphism

A alg ⌦ k k n ! B alg ⌦ k k n .
Therefore, by the universal property of CAlg ad k (X) as a pullback we conclude that the ( ) alg : fCAlg k (X) ! CAlg k (X) can be upgraded to a functor

( ) ad : fCAlg k (X) ! CAlg ad k (X), as desired. Proposition 3.3.1.5. Both ( ) alg : Top R (T ad (k )) ! Top R (T disc (k )) and ( ) sh : Top R (T ad (k )) ! Top R (T ét (k )) admit right adjoints L : Top R (T disc (k )) ! Top R (T ad (k )) L sh : Top R (T ét (k )) ! Top R (T ad (k ))).
Proof. This is an immediate consequence of [Lur11c, Theorem 2.1].

We now proceed to have a better understanding of the action of L at the level of T disc -structures:

Construction 3.3.1.6. Let (X, O) 2 Top R (T ad (k )
) be a T ad (k )-structured 1-topos. Consider the comma 1-category fCAlg k (X) /O which is a presentable 1-category thanks to [Por15a, Corollary 9.4]. The underlying algebra functor induces a well defined functor at the level of comma 1-categories:

( ) alg : fCAlg k (X) /O ! CAlg k (X) /O alg .
Thanks to [Por15a, Corollary 9.5] the above functor commutes with limits and sifted colimits. Thanks to the Adjoint functor theorem it follows that ( ) alg : fCAlg k (X) /O ! CAlg k (X) /O alg admits a left adjoint which we shall denote X : CAlg k (X) /O alg ! fCAlg k (X) /O , or simply if the underlying 1-topos X is made explicit.

We refer the reader to [Lur16, §7.3] for the notion of t-completeness of modules. Construction 3.3.1.7. Let A 2 CAlg k (X) /O alg be a T disc -structure on X. We define A n as the pushout of the diagram

A[u] A A A n u7 !t n u7 !0 (3.3.1.1) in the 1-category CAlg k (X) /O alg n .
Where O alg n is defined in a similar way and A[u] denotes the free algebra on one generator in degree 0 over A. As is a left adjoint we obtain a pushout square

(A[u]) (A) (A) (A n ) u7 !t n u7 !0 (3.3.1.2)
in the 1-category fCAlg k (X) On . Moreover, as in an 1-topos every epimorphism is effective, and preserves epimorphisms, see Theorem .

(A[u]) alg (A) alg (A) alg (A n ) alg u7 !t n u7 !0 in the 1-category CAlg k (X) /O alg n .
Therefore, for each integer n 1 the unit of the adjunction ( , ( ) alg ) induces morphisms

f A,n : A n ! (A) alg n such that the ideal I n := ker (⇡ 0 (A) ! ⇡ 0 (A n )) is sent to the ideal J n := ker ⇡ 0 (A) alg ! ⇡ 0 (A) alg n .
Therefore, the universal property of t-completion induces a canonical morphism

f A : A t ! (A) alg ,
in the 1-category CAlg k (X). We can now understand explicitly the composite ( ) alg :

Proposition 3.3.1.11. Let (X, O) 2 Top R (T disc (k )
) such that the underlying 1-topos X has enough geometric points and O is stricly Henselian. Let A 2 CAlg k (X) /O be an almost of finite presentation T disc (k )-structure on X which we assume further to be strictly Henselian. Then the canonical map

f A : A t ! (A)
alg introduced in Theorem 3.3.1.7 is an equivalence in the 1-category CAlg ad k (X) / (O) alg . Proof. We wish to show that the natural map

f A : A t ! (A) alg constructed in Theorem 3.3.1.7 is an equivalence whenever A 2 CAlg k (X) /O alg is almost of finite presentation.
By hypothesis X has enough geometric points. Thus in order to show that f A is an equivalence it suffices to show that its inverse image under any geometric point (x 1 , x ⇤ ) : X ! S, x 1 f A , is an equivalence in the 1category CAlg k . Set A := x 1 A. Thanks to [Por15c, Theorem 1.12] we deduce that S (A) alg ' x 1 (A) alg . We are thus reduced to the case where X = S.

The 1-category (CAlg k ) /O alg is generated under sifted colimits by free objects of the form {k [T 1 , . . . , T m ]} m 1 . Thanks to Theorem .3.0.1 we conclude that (fCAlg k ) /O := (fCAlg k ) /O (S) is generated under sifted colimits by the family

{ (k [T 1 , . . . T m ])} m . As A 2 (CAlg k ) /x 1 O alg is
almost of finite presentation we conclude that it can be written as a retract of a filtered colimit of a diagram of the form

A 0 ! A 1 ! A 2 ! . . . ,
where A 0 is an ordinary commutative ring of finite presentation over k and A i+1 can be obtained from A i as the following pushout

k [S n ] k [X] A i A i+1 , (3.3.1.3)
where k [S n ] is the free simplicial k -algebra generated in degree n by a single generator. Notice that, since A is almost of finite presentation we can choose the above diagram in such a way that for i > 0 sufficiently large, we have surjections ⇡ 0 (A i ) ! ⇡ 0 (A i+1 ). As is a left adjoint it commutes, in particular, with pushout diagrams. We conclude that the diagram

(k [S n ]) (k [X]) (A i ) (A i+1 ), (3.3.1.4) is a pushout diagram in the 1-category fCAlg k (X) /O and the morphism (A i ) ! (A i+1 ) is moreover an epimorphism on ⇡ 0 . For each n > 0, the morphism k [S n ] ! k [X] is an effective epimorphism. As is a left adjoint, the morphism (k [S n ]) ! (k [X]
) is an epimorphism in the (hypercomplete) 1-topos X and thus an effective epimorphism. Thanks to [PY16a, Proposition 3.14] it follows that the morphism

(k [S n ]) alg ! (k [X]) alg
is an effective epimorphism. Therefore, as the transformation of pregeometries ✓ :

T ét (k ) ! T ad (k ) is unrami- fied, see Appendix B, [Lur11a, Propositon 10.3] implies that the diagram, (k [S n ]) alg (k [X]) alg (A i ) alg (A i+1 ) alg , (3.3.1.5)
is a pushout square in CAlg k . By induction we might assume that

(A i ) alg is equivalent to (A i ) t .
The transformation of pregeometries ( ) t : T ét (k ) ! T ad (k ) is given by t-completion along the (t)-locus. Therefore, one has a canonical equivalence

(k [X]) alg ' k hXi sh ,
where latter the t-completion of the strictly Henselianization of k [X]. We claim that the natural map

(k [S n ]) alg ! k [S n ] sh t is an equivalence: notice that k[S n ] fits into a pushout diagram k [S n 1 ] k [X] k [X] k [S n ],
the result then follows by induction on n 0 and the case n = 0 was already treated. Since

(A i ) alg ! (A i+1 ) alg is surjective on ⇡ 0 , it follows that ⇡ 0 ( (A i+1 ) alg ) is t-complete. For each i 0 the ⇡ 0 (A i+1 ) alg -modules ⇡ n (A i+1 ) alg are of finite presentation, thus t-complete ⇡ 0 ( (A i+1 ) alg )-modules. It follows that (A i+1 ) alg is t-complete by [Lur16, Theorem 7.3.4.1].
Let A i+1 ! B be a morphism in CAlg k whose target is strictly Henselian and t-complete. Thanks to (3.3.2) such morphism induces morphisms

A i ! B and k [T ] ! B compatible with both k [S n ] ! k [T ] and k [S n
] ! A i , in the 1-category CAlg k . By induction the effect of ( )

on A i , k [S n
] and k [X] agrees with strictly henselianization followed by t-completion. As B is both strictly henselian and t-complete it follows that the map A i+1 ! B induces a well defined morphism from the diagram displayed in (3.3.1.5) to B. It follows that (A i+1 ) alg satisfies the universal property of t-completion for the derived k -algebra A i+1 . As (A i+1 ) alg is t-complete we conclude that the morphism

f Ai+1 : (A sh i+1 ) t ! (A i+1 ) alg ,
where A sh i+1 denotes the strict henselianization of A i+1 , is necessarily an equivalence. Let A := colim i A i in the 1-category CAlg k . Fix i 0, then ⌧ i (A) alg ' ⌧ i (A j ) alg for j sufficiently large. We conclude then that

⇡ i ( (A) alg ) is t-complete for i 0. [Lur16, Theorem 7.3.4.1] implies that (A) alg is t-complete.
Reasoning as before we conclude that it satisfies the universal property of t-completion with respect to A. It follows that

f A : A t ! (A) alg
is an equivalence in the 1-category CAlg k , the result now follows.

Warning 3.3.1.12. The functor ( ) alg is not in general equivalent to the t-completion functor ( ) t . In fact, both ( ) alg and commute with filtered colimits, thus also their composite ( ) alg

. Therefore, the composite ( ) alg commutes with filtered colimits which is not the case of the t-completion functor, in general.

We will need also the following ingredient:

Construction 3.3.1.13. Denote by k n the reduction of k modulo (t n ). Reduction modulo (t n
) induces a transformation of pregeometries

p n : T ad (k ) ! T disc (k n ) Spf R 7 ! Spec R n where R n := R ⌦ k k n . Precomposition along p n induces a morphism at the level of structured 1-topoi p 1 n : Top R (T disc (k n )) ! Top R (T ad (k ))
which is given on objects by the formula

(X, O) 2 Top R (T disc (k )) 7 ! (X, O p n ) 2 Top R (T ad (k )).
Given X 2 Top R we obtain an induced functor at the level of structures p 1 n : CAlg k n (X) ! fCAlg k (X) given on objects by the formula

O 2 CAlg k n (X) 7 ! p 1 n O := O p n 2 fCAlg k (X).
Notice that, we have a commutative triangle of the transformations of pregeometries of the form

T disc (k ) T ad (k ) T disc (k n ) ( ) t ⌦ k k n pn .
For this reason, for every X 2 Top R , it follows that the composite ( ) alg p 1 n : CAlg k n (X) ! CAlg k (X) coincides with the usual forgetful functor CAlg k n (X) ! CAlg k (X) along the induced map of derived rings k ! k n . Notice that the latter functor admits a left adjoint which is given by extension of scalars along k ! k n , i.e. it is given on objects by the formula

O 2 CAlg k (X) 7 ! O ⌦ k k n 2 CAlg k (X)
Notation 3.3.1.14. We will denote by ( ) n : CAlg k (X) ! CAlg k n (X) the functor given by extension of scalars along the canonical morphism of derived rings k ! k n :

O 2 CAlg k (X) 7 ! O n := O ⌦ k k n 2 CAlg k (X) It follows by [Lur11c, Theorem 2.1] that p 1 n admits a right adjoint L n : Top R (T ad (k )) ! Top R (T disc (k n )) which we can explicitly describe: Proposition 3.3.1.15. The functor p 1 n : Top R (T disc (k n )) ! Top R (T ad (k )) admits a right adjoint L n : Top R (T ad (k )) ! Top R (T disc (k n ))
whose restriction to the full subcategory of Top R (T ad (k )) spanned by those couples (X, O) such that the underlying 1-topos X has enough points is given on objects by the formula

(X, O) 2 Top R (T ad (k )) 7 ! (X, O alg n ) 2 Top R (T disc (k n )).
Proof. The existence of a left adjoint

L n : Top R (T ad (k )) ! Top R (T disc (k n )) follows directly from [Lur11c, Theorem 2.1]. Let (X, O) 2 Top R (T disc (k n )) and (Y, O 0 ) 2 Top R (T ad (k )) be such that X 2 Top R has enough points. Given any geometric morphism (f 1 , f ⇤ ) : X ! Y
we have a morphism of fiber sequences of the form

Map fCAlg k (X) f 1 O 0 , p 1 n O Map Top R (T ad (k )) (X, p 1 n O), (Y, O 0 ) Map Top R (X, Y) Map CAlg k n (X) (f 1 O 0 ) alg n , O Map Top R (T disc (k n )) (X, O), (Y, (O 0 ) alg n ) Map Top R (X, Y) q p (3.3.1.6) where q : Map fCAlg k (X) f 1 O 0 , p 1 n , O ! Map CAlg k n (X) (f 1 O 0 ) alg n , O coincides with the composite Map fCAlg k (X) f 1 O 0 , p 1 n O Map CAlg k (X) (f 1 O 0 ) alg , p 1 n O alg Map CAlg k n (X) (f 1 O 0 ) alg n , O . (-) alg
In order to prove the assertion of the proposition it suffices to show that the morphism p displayed in (3.3.1.6) is an equivalence of mapping spaces. Thanks to the fact that the horizontal arrow diagrams in (3.3.1.6) form fiber sequences we are reduced to prove that q is an equivalence of mapping spaces. As X has enough points we reduce ourselves to prove the statement of the Theorem at the level of stalks. For this reason we can assume from the start that X = S. Both target and source of q commute with filtered colimits on the first argument, thus we are reduced, as in the proof of Theorem 3.3.1.11 to prove that q is an equivalence whenever

f 1 O 0 ' (k [T 1 , . . . , T n ]).
We have natural equivalences of mapping spaces

Map fCAlg k (k [T 1 , . . . T m ]), p 1 n O ' Map CAlg k k [T 1 , . . . T m ], (p 1 n O) alg ' Map CAlg k n k [T 1 , . . . T m ] n , (p 1 n O) alg ' Map CAlg k n (k n [T 1 , . . . T m ], O) .
The result now follows from the observation that

(k [T 1 , . . . T m ]) alg n ' k n [T 1 , . . . , T m ] in the 1-category CAlg k n , which is a direct consequence Theorem 3.3.1.11. Corollary 3.3.1.16. Let X 2 Top R be an 1-topos. The functor L n : Top R (T ad (k )) ! Top R (T disc (k n ))
introduced in Theorem 3.3.1.15 induces a well defined functor at the level of the corresponding 1-categories of structures

( ) ad n : fCAlg k (X) ! CAlg k n (X), given on objects by the formula O 2 fCAlg k (X) 7 ! O alg n 2 CAlg k n (X).
Moreover, the functor ( ) ad n is a left adjoint to the forgetful

p 1 n : CAlg k n (X) ! fCAlg k (X)
Proof. The existence of ( ) ad n is guaranteed by Theorem 3.3.1.15. The fact that ( ) ad n is a left adjoint to

p 1 n : CAlg k n (X) ! fCAlg k (X)
follows from the proof of Theorem 3.3.1.15 together with the fact that both ( ) ad n and p 1 n are defined at the level of 1-categories of structures on the same underlying 1-topos. Notation 3.3.1.17. Consider the forgetful functor Top

R (T disc (k n )) ! Top R (T disc (k ))
given by restriction of scalars along the morphism k ! k n . We will denote

⇥ Spec k Spec k n : : Top R (T disc (k )) ! Top R (T disc (k n )) its right adjoint. Corollary 3.3.1.18. For each n 1, the composite L n L : Top R (T disc (k )) ! Top R (T disc (k n )) coincides with the base change functor ⇥ Spec k Spec k n : Top R (T disc (k )) ! Top R (T disc (k )), (X, O) 2 Top R (T disc (k )) 7 ! (X, O) ⇥ Spec k Spec k n 2 Top R (T disc (k n ))
Proof. This is a direct consequence of the definitions together with the commutative triangle displayed in Theorem 3.3.1.13.

Comparison with derived formal geometry

Our main goal now is to give comparison statements between T ad (k )-structured 1-topoi and locally adic ringed 1-topoi. The latter corresponding to couples (X, O) where O is a CAlg ad k -valued sheaf on the 1-topos X. We shall moreover fix a couple (X, O) 2 fCAlg k (X) /O throughout this whole §. Definition 3.3.2.1. Let X be an 1-topos and A 2 CAlg ad k (X) be a CAlg ad k -valued sheaf on X. We say that A is topologically almost of finite presentation if A is t-complete, the sheaf ⇡ 0 (A) is topologically finitely generated and for each i > 0 the homotopy sheaf ⇡ i (A) is finitely generated as a ⇡ 0 (A)-module. Definition 3.3.2.2. Let X be an 1-topos and consider the functor ( ) ad : fCAlg k (X) ! CAlg ad k (X) introduced in Theorem 3.3.1.4. We say that A 2 fCAlg k (X) is topologically almost of finite presentation if the underlying sheaf of adic algebras A ad is topologically almost of finite presentation. We denote fCAlg taft k (X) the 1-category of topologically almost of finite presentation local T ad (k )-structures on X.

Construction 3.3.2.3. Consider the adjunction , ( ) alg : CAlg k (X) /O alg ! fCAlg k (X) /O of Theorem 6.2.3.5 and let ( ) disc : CAlg ad k (X) ! CAlg k (X)
denote the canonical functor obtained by forgetting the adic structure. Then the couple

( ) disc , ( ) ad : CAlg ad k (X) /O ad ! fCAlg k (X) /O
forms an adjunction pair after restriction ad , ( )

ad := ( ) disc , ( ) ad : CAlg ad,taft k (X) /O ad ! fCAlg taft k (X) /O ,
where CAlg ad,taft k (X) /O ad denotes the full subcategory of CAlg ad k (X) /O ad spanned by those objects A 2 CAlg ad k (X) topologically almost of finite presentation.

In order to see this consider the unit id ! ( ) alg of the adjunction in Theorem 6.2.3.5. It follows by the construction of ( ) ad : fCAlg k (X) ! CAlg ad k (X) that we have an equivalence

( ) alg ' ( ) disc ( ) ad in the 1-category Fun fCAlg k (X) /O , CAlg k (X) /O alg . Therefore, for each A 2 CAlg ad,taft k (X) /O ad the unit of adjunction A disc ! (A disc )
alg induces a canonically defined, up to a contractible space of choices, morphism

A ' A t ! ad (A) ad .
This construction is functorial and thanks to our previous considerations it satisfies the universal property of a unit of adjunction. Therefore we obtain an adjunction ad , ( )

ad : CAlg ad,taft k (X) /O ad ! fCAlg taft k (X) /O , as desired.
Theorem 3.3.2.4. Let X be an 1-topos with enough geometric points. Consider the functor

( ) ad : fCAlg k (X) /O ! CAlg ad k (X) /O ad introduced in Theorem 3.3.1.4.
Then the induced restriction functor

( ) ad : fCAlg taft k (X) /O ! CAlg ad k (X) /
: CAlg ad,taft k (X) /O ad ! fCAlg k (X) /O
is fully faithfully when restricted to the full subcategory C. [PY16a, Lemma 3.13] implies that the right adjoint functor ( ) ad is conservative, the conclusion now follows.

Remark 3.3.2.5. Theorem 3.3.2.4 can be interpreted as a rectification statement. Indeed, an element A 2 fCAlg k (X) corresponds to a functor A : T ad (k ) ! X satisfying the axioms for a T ad (k )-structure on X. Morphisms in A ! B in fCAlg k (X) correspond to local morphisms in Fun (T ad (k ), X).

On the other hand, objects in the 1-category CAlg ad k (X) correspond to a derived k -algebra on X, A 2 CAlg(X), together with the given of an adic topology on the sheaf of ordinary k -algebras, ⇡ 0 (A). Moreover, morphisms in CAlg ad k (X) correspond to morphisms on CAlg(X) which are continuous adic morphisms at the level of ⇡ 0 . Therefore, a priori, one could expect that specifying morphisms in the 1-category fCAlg taft k (X) would require an increase amount of higher coherence data when compared to the adic case.

Construction 3.3.2.6 (The Spf-construction). Let A 2 CAlg ad k be a derived adic k -algebra. We can asso- ciate to A an object Spf A := (X A , O A ) 2 Top R (T ad (k )) as follows: we let X A := HShv ad A 2 Top R de- note the hypercompletion of the 1-topos Shv ad A introduced in [Lur16, Notation 8.1.1.8]. We define moreover O A : T ad (k ) ! X A as the T ad (k )-structure on X A determined by the formula Spf(R) 2 T ad (k ) 7 ! ⇣ B 2 CAlg ad,ét A 7 ! Map CAlg ad k (R, B)

⌘

where CAlg ad,ét A denotes the full subcategory of CAlg ad A spanned by those derived A-algebras B étale over A. One checks directly that

O A : T ad (k ) ! X A is indeed a T ad (k )-structure on X A . Such association is functorial in A 2 CAlg ad
k and we thus obtain a well defined functor (up to contractible space of choices) Spf :

⇣ CAlg ad k ⌘ op ! Top R (T ad (k )),
which we refer as the Spf-construction functor.

L n : Top R (T ad (k )) ! Top R (T disc (k n )) introduced in Theorem 3.3.1.14. Given A 2 CAlg ad k , it follows that L n (Spf(A)) ' (X A , O A,n ) where O A,n := O alg A ⌦ k k n 2 CAlg k (X). Proposition 3.3.2.9. The functor Spf : CAlg ad k op ! Top R (T ad (k )
) is fully faithful. Moreover, its essentially image corresponds precisely to the full subcategory of Top

R (T ad (k )) spanned by those couples (X, O) 2 Top R (T ad (k )) such that (X, O alg ) 2 Top R (T disc (k )) is equivalent to a formal spectrum. Proof. Let A, B 2 CAlg ad k and consider the corresponding formal spectrums Spf(A) and Spf(B) 2 Top R (T ad (k )). The datum of a morphism of local T ad (k )-structures f : Spf(A) ! Spf(B) is equivalent to the datum of a ge- ometric morphism of 1-topoi (f 1 , f ⇤ ) : X A ! Y B together with a natural transformation ↵ : f 1 O B ! O A .
Applying the underlying algebra functor at the level of structures we obtain a morphism

↵ alg : f 1 O ad B alg ! O ad A alg in the 1-category fCAlg k (X A ).
The unit of the adjunction (f 1 , f ⇤ ) produces a well defined morphism of derived k -algebras : B ! A, up to contractible indeterminacy.

By the construction of the underlying 1-topoi of both Spf(A) and Spf(B) together with [Lur16, Remark 8.1.1.7] it follows that the morphism : B ! A is continuous with respect to the adic topologies for both A and B. We obtain thus a well defined morphism of mapping spaces

: Map Top R (T ad (k )) (Spf A, Spf B) ! Map CAlg ad k (B, A) .
Let : B ! A be a continuous morphism of derived adic k -algebras. In order to show that the functor

Spf : CAlg ad k op ! Top R (T ad (k ))
is fully faithful it suffices to show that the fiber Z := fib ( ) is contractible for any such . To any continuous adic morphism, we can attach a well defined, up to contractible indeterminacy, morphism on the corresponding (formal) étale sites. We have thus a canonical morphism at the level of mapping spaces

✓ : Map CAlg ad k (B, A) ! Map Top R (X A , Y B ) . Let (f 1 , f ⇤ ) : X A ! Y B
be a morphism of 1-topoi such that it lies in the essential image of under ✓. The fiber over (f 1 , f ⇤ ) induces a fiber sequence of mapping spaces:

Map fCAlg k (X) f 1 O B , O A Map Top R (T ad (k )) (Spf A, Spf B) Map Top R (X A , Y B ) . ✓ Consider the commutative diagram in the 1-category S Z Map fCAlg k (X) f 1 O B , O A Map Top R (T ad (k )) (Spf(A), Spf(B)) { } W Map CAlg ad k (B, A) {(f 1 , f ⇤ )} Map Top R (X A , Y B ) (3.3.2.1)
where both the upper rectangle and and the bottom right square are pullback diagrams. It follows then that we can identify Z with the pullback , f ⇤ ) is also uniquely determined by , up to a contractible space of choices. As a consequence we can identify Z with the fiber product:

Z ' Map fCAlg k (X) f 1 O B , O A ⇥ W { }. Let F : Spf(A) ! Spf(B)
Z ' Map fCAlg k (X) f 1 O B , O A ⇥ W { }.
We have a sequence of equivalences of mapping spaces

Map fCAlg k (X) f 1 O B , O A ⇥ W { } ' Map fCAlg k (X) f 1 O B , lim n 1 (O A,n ) ⇥ W { } ' ⇣ lim n 1 Map fCAlg k (X) f 1 O B , O A,n ⌘ ⇥ W { }
We can further identify the last term with

⇣ lim n 1 Map fCAlg k (X) f 1 O B , O A,n ⌘ ⇥ W { } ' ⇣ lim n 1 Map CAlg k n (X) f 1 O B,n , O A,n ⌘ ⇥ W { }
For each n 1, denote n the base change of to k n . Passing to the limit over n 1 we can further identify the last term with

lim n 1 ⇣ Map CAlg k n (X) f 1 O B,n , O A,n ⇥ Wn { n } ⌘ ' lim n 1 ⇣ Map Top R (T disc (k n )) (A n , B n ) ⇥ Wn { n } ⌘ , (3.3.2.2)
where W n is defined as the fiber product of the corresponding diagram obtained as the reduction modulo t n of the bottom right square, displayed in (3.3.2.1). Thanks to the proof of [Lur16, Corollary 1.2.3.5.] each term in displayed limit displayed in (3.3.2.2) can be identified with

Map CAlg k n (B n , A n ) ⇥ CAlg k n (Bn,An) { n }
which is thus a contractible space. The result now follows by a simple analysis on the corresponding Milnor exact fiber sequence. Definition 3.3.2.12. We denote by dfSch the full subcategory of dfDM spanned by those objects

X = (X, O) such that (X, ⇡ 0 O alg
) is equivalent to an ordinary derived formal scheme over k . We refer to objects in dfSch as derived k -adic formal schemes. We also define the 1-category of topological almost of finite presentation derived k -adic schemes as dfSch taft := dfDM taft \ dfSch. Remark 3.3.2.14 (Global spectrum construction). Let fDM k denote the category of ordinary Deligne-Mumford stacks and let X 2 fDM k be a Deligne-Mumford stack. To X we can associate a T ad (k )-structured 1-topos as follows: Let X fét denote the formally étale site on X. Denote by X := Shv ét (X) ^the hypercompletion of the 1-topos Shv(X fét , T fét ). We define a T ad (k )-structure on X by the formula

Spf(R) 2 T ad (k ) 7 ! Y 2 X fét 7 ! Map fDM k (Y, Spf(R)) 2 S .
In this case, O(A 1 k ) 2 CAlg k (X) corresponds to the usual structure sheaf of continuous adic functions on X. This association is functorial and it provides us with a fully faithful embedding

fDM k ✓ dfDM k of 1-categories. 3.3.3 Derived 1-categories of modules for T ad (k )-structured spaces Definition 3.3.3.1. Let X := (X, O) 2 Top R (T ad (k )).
We define the 1-category of modules on X as

Mod O := Sp Ab fCAlg k (X) /O ,
where Ab denotes the abelianization functor, see [Lur12c, §1.4], and Sp the stabilization functor.

Remark 3.3.3.2. Let (X, O) be as above. The 1-category Mod O is stable. Construction 3.3.3.3. Given (X, O) 2 Top R (T ad (k ))
we can also consider the 1-category of modules on its algebraization (X, O alg

) defined as Mod O alg := Shv D(Ab) (X), where D(Ab) := Mod Z denotes the derived 1-category of Z-modules. Thanks to [Lur12c,Theorem 7.3.4.13] one has a natural equivalence

Mod O alg ' Sp (Ab (CAlg k (X) O alg )) , in the 1-category Cat st 1 .
As the underlying algebra functor ( ) alg : fCAlg k (X) /O ! CAlg k (X) /O alg is a right adjoint it induces an exact functor at the level of derived 1-categories of modules denoted

g alg : Mod O ! Mod O alg .
We claim that the left adjoint : CAlg k (X) /O alg ! fCAlg k (X) induces also a well defined functor

f ad : Mod O alg ! Mod O ,
which is a left adjoint to g alg . It suffices to prove that commutes with finite limits. We start by observing that as the composite ( ) alg agrees with the t-completion functor on the full subcategory of almost of finite presentation objects CAlg k (X) afp /O it commutes with small limits on CAlg k (X) afp /O alg . As ( ) alg is a conservative right adjoint, it follows that itself commutes with finite limits on CAlg k (X) afp /O alg . Let now A 2 CAlg k (X) /O alg be a general object. We can realize A as a filtered colimit of almost of finite presentation objects in CAlg k (X) /O alg . Let {A i } i be a diagram indexed by a finite 1-category I, and for each i 2 I choose a presentation

A i ' colim m2J A i,m ,
where A i,m is almost of finite presentation and J is a filtered 1-category. We have thus a sequence of equivalences in the

1-category CAlg k (X) /O alg (lim i A i ) alg ' colim m (lim i A i,m ) alg ' colim m lim i (A i,m ) t ' lim i colim m (A i,m ) t ' lim i colim m (A i,m ) alg ' lim i (A i,m ) alg
and the conclusion now follows as in the preceding case. ) alg ' O alg . Then the functor

g alg : Mod O ! Mod O alg is an equivalence of stable 1-categories.
Proof. Let f ad : Mod O alg ! Mod O denote a left adjoint to g alg which is induced by the functor introduced in §3.1. We want to show that f ad is an inverse to g alg , as functors. Notice that the functor g alg is conservative as ( ) alg was already conservative. Therefore, we are reduced to show that f ad is a fully faithfully functor. It suffices to show that the unit ⌘ of the adjunction (f ad , g alg ) is an equivalence. As X has enough geometric points we reduce ourselves to check the last assertion at the level of stalks. We are thus reduced to the case X = S. In this case, the 1-category Mod O alg is compactly generated by O alg 2 Mod O alg . The ( ) alg commutes with filtered colimits (even sifted colimits) thus we deduce that also g alg commutes with filtered colimits. As g alg is an exact functor between stable 1-categories we conclude that it commutes with all colimits. Therefore, the unit ⌘ commutes with colimits. We are thus reduced to check that ⌘ is an equivalence on the compact generator O alg 2 Mod O alg . By our assumption on O alg the result follows thanks to Theorem 3.3.1.11. )-modules and vanish for sufficiently small i.

k -adic cotangent complex

In this §we will introduce the notion of formal cotangent complex, which will prove to be of fundamental importance to us: we have a projection functor

⌦ 1 ad : Mod O ! fCAlg k (X) /O
, which is given by evaluation on the object (S 0 , ⇤) 2 S fin ⇤ ⇥ T Ab . The functor ⌦ 1 ad admits a left adjoint ⌃ 1 ad : fCAlg k (X) /O ! Mod O . We refer the reader to [PY17a, §5.1] and [Lur12c, §7.5] for more details about these constructions. Proof. The proof is a direct consequence of the existence of a left adjoint

⌃ 1 ad : fCAlg k (X) /O ! Mod O . Set L ad O/A := ⌃ 1 ad (O ⌦ A O).
For every M 2 Mod O we have a sequence of natural equivalences of mapping spaces of the form

Der ad A (O, M) ' Map fCAlg k (X) A//O (O, O M ) ' Map fCAlg k (X) A//O (O, ⌦ 1 a d(M )) ' Map fCAlg k (X) O//O (O ⌦ A O, O M ) ' Map Mod O (⌃ 1 ad (O ⌦ A O), O) ' Map Mod O ⇣ L ad O/A , M ⌘ ,
and the result follows.

Mod O Mod O alg fCAlg k (X) /O CAlg k (X) /O alg , g alg ⌦ 1 ad ⌦ 1
( ) alg therefore passing to left adjoints we obtain a commutative diagram

Mod O Mod O alg fCAlg k (X) /O CAlg k (X) /O f ad ⌃ 1 ad ⌃ 1 (3.3.4.1)
in the 1-category Cat 1 . The commutative of (3.3.4.1) provide us with a natural map

f ad L B alg /A alg ! L ad B/A
in the 1-category Mod O .

Proposition 3.3.4.6. Let A ! B be a morphism in fCAlg k (X) /O and consider the algebraic cotangent complex L B alg /A alg associated to the morphism A alg ! B alg . Then the natural map introduced in Theorem 3.3.4.5

f ad L B alg /A alg ' L ad B/A is an equivalence in the 1-category Mod O .
Proof. The construction of the adic cotangent complex commutes with filtered colimits of local T ad (k )-structures.

Therefore we can suppose that the morphism A ! B is topologically almost of finite presentation and A and B and A is itself topologically almost of finite presentation. Let A ! B be a morphism in fCAlg k (X) /O and consider L B alg /A alg 2 Mod O alg the algebraic cotangent complex associated to A alg ! B alg . By applying the functor f ad : Mod O alg ! Mod O , we obtain the following sequence of equivalences of mapping spaces

Map Mod O f ad (L B alg /A alg ), M ' Map Mod O alg L B alg /A alg , g alg (M ) ' Map CAlg k (X) /O alg A alg , A alg g alg (M ) ' Map CAlg k (X) /O alg A alg , (A M ) alg ' Map fCAlg k (X) /O (A, A M )
where the latter equivalence holds by fully faithfulness of the functor ( ) alg , as A is topologically almost of finite presentation and t-complete and whenever M is coherent, which we can assume from the start thanks to Theorem 3.3.4.4.

Proposition 3.3.4.7. Let f : A ! B and g : B ! C be morphisms in the 1-category fCAlg k (X) /O . Then one has a fiber sequence

L ad B/A ⌦ B C ! L ad C/A ! L ad C/B in Mod O .
Proof. This is a direct consequence of [PY17a, Proposition 5.10].

L ad B/A ⌦ B D ! L ad D/C
is an equivalence in the 1-category Mod O .

Proof. The assertion is a particular case of [PY17a, Proposition 5.12].

Postnikov towers of k -adic spaces

Definition 3.3.5.1. Let X = (X, O) 2 Top R (T ad (k )) and M 2 (Mod O ) 1 be an O-module concentrated in homological degrees 1. A k -adic square zero extension of X by M consists of a T ad (k )-adic structured 1-topos X 0 = (X, O 0
) equipped with a morphism f : X ! X 0 satisfying: (i) The underlying geometric morphism of f is equivalent to the identity of X;

(ii) There exists an k -adic derivation

d : L ad X ! M [1] 2 Mod O such that we have a pullback diagram in the 1-category fCAlg k (X) /O O 0 O O O M [1] d d0
where d 0 denotes the trivial k -adic derivation.

Definition 3.3.5.2. Let T be a pregeometry and let n 1 be an integer. We say that T is compatible with n-truncation if for every 1-topos X, every T-structure O : T ! X and every admissible morphism U ! V in T, the induced square

O(U ) ⌧ n O(U ) O(V ) ⌧ n O(U ) is a pullback diagram in X.
O(U ) ⌧ 0 O(U ) O(U ) ⌧ 0 (V ) (3.3
B 0 such that B ⇠ = (B 0 ) t .
The morphism U ! V being admissible in T ad (k ) implies that the induced morphism B ! A is formally étale. [dJ + , Tag A0R1, Lemma 7.9.10.3] implies that the morphism B ! A can be realized as the t-completion of k -algebras B 0 ! A 0 , where A 0 is an étale k [T 1 , . . . , T n ] itself. Therefore, the morphism of spaces

O(U ) ! O(V ) can be identified with a morphism O sh (Spec A 0 ) ! O sh (Spec B 0 )
and similaryly for the morphism

⌧ 0 O(U ) ! T 0 O(V ). Therefore we can identify the diagram (3.3.5.1) with O sh (Spec A 0 ) ⌧ 0 O sh (Spec A 0 ) O sh (Spec B 0 ) ⌧ 0 O sh (Spec B 0 )
in the 1-category S. The result now follows thanks to [Lur11c, Proposition 4.3.28].

Definition 3.3.5.7. Let X = (X, O) 2 Top R (T ad (k )). We define its n-th truncation t n (X) := (X, ⌧ n O) 2 Top R (T ad (k )). Proposition 3.3.5.8. Let X = (X, O) 2 Top R (T ad (k )).
Then for each integer n 0, the n + 1-th truncation t n+1 (X) is a square zero extension of t n (X). In particular, when X is a derived k -adic Deligne-Mumford stack then for each n 0 the n-truncation t n (X) is again a derived k -adic Deligne-Mumford stack.

Proof. We have a canonical morphism t n (X) ,! t n+1 (X) induced by the identity functor on the underlying 1-topos X and the natural map ⌧ n+1 O ! ⌧ n O at the level of structures. Let B := ⌧ n+1 O and A := ⌧ n O. Thanks to [Lur12c,Corollary 7.4.1.28] we deduce that the induced morphism at the level of underlying algebras B alg ! A alg is a square zero extension. Thus we can identify B alg with the pullback of the diagram

B alg A alg A alg A alg L B alg /A alg d d0 (3.3.5.2) in the 1-category CAlg k (X) /⌧ n O . Consider the induced k -adic derivation f ad (d) : L ad A ! L ad B/A
and form the pullback diagram

B 0 A A A L ad B/A d0 (3.3.5.3)
in the 1-category fCAlg k (X) /A . In this way the canonical morphism B 0 ! A is a k -adic square zero extension and we have a canonical map B ! B 0 . As filtered colimits commute with finite limits we reduce ourselves to the case that O, and therefore both A and B, are topologically almost of finite presentation. Thanks to Theorem 3.3.1.11, the functor applied to the pullback diagram (6.3.2.12) is the identity. Thus by conservativity of ( ) alg it follows that the diagram

(B alg ) (A alg ) (A alg ) (A alg L B alg /A alg ) d d0 (

Derived rigidification functor 3.4.1 Construction of the rigidification functor

Raynaud's generic fiber construction [Bos05, §8], induces a transformation of pregeometries

( ) rig : T ad (k ) ! T an (k),
which is moreover a localization of categories with respect to those morphisms Spf(A) ! Spf(B) such that there exists there exists a k -adic complete algebra C together with continuous adic morphisms C ! A and C ! B such that they induce equivalence after inverting t,

A ⌦ k k ' C ⌦ k k ' B ⌦ k k. Proposition 3.4.1.1. Precomposition along the transformation of pregeometries ( ) rig : T ad (k ) ! T an (k) in- duces a functor ( ) + : Top R (T an (k)) ! Top R (T ad (k )),
which admits a right adjoint denoted

( ) rig : Top R (T ad (k )) ! Top R (T an (k)).
referred to as the (derived) rigidification functor. These considerations imply the following useful result:

Proof. It is a direct consequence of [Lur11c, Theorem 2.1]. Lemma 3.4.1.2. For each integer n 0, we have a commutative diagram Top R (T an (k)) Top R (T ad (k )) Top R (T an (k)) n Top R (T ad (k )) n ( ) + ( ) +
Corollary 3.4.1.4. Let X = (X, O) be a T ad (k )-structured space which is equivalent to an ordinary k -adic formal scheme topologically of finite presentation. Then X rig is equivalent to an ordinary k-analytic space which agrees with the usual generic fiber of X.

Proof. The question is local on X. We can thus assume that X ' Spf(A), where A 2 CAlg ad k , where A is a topologically of finite presentation ordinary k -adic algebra. Therefore, choosing generators and relations for A we can find an (underived) pullback diagram of the form

Spf(A) A m k Spf(k ) A n k (3.4.
Spf(k ) rig ' Sp k, (A m k ) rig ' A m k , (A n k ) rig ' A n k .
As ( ) rig is a right adjoint, it commutes with pullback diagrams. We thus have a pullback diagram in the 1-

category Top R (T an (k)) Z rig A m k Sp(k) A n k Theorem 3.4.1.3 implies that t 0 Z rig ' t 0 (Z) rig . As t 0 (Z) ' Spf(A) we deduce that (Spf(A)) rig is equivalent to the (underived) pullback diagram Spf(A) rig A m k Sp(k) A n k
computed in the category of rigid k-analytic spaces. This is precisely the usual generic fiber construction applied to Spf A.

Lemma 3.4.1.5. Let f : Z ! X be a closed immersion of derived k -adic Deligne-Mumford stacks topologically almost of finite presentation. Then f rig is a closed immersion in the 1-category dAn.

Proof. It suffices to show that the truncation t 0 (f rig

) : t 0 (Z rig ) ! t 0 (X rig
) is a closed immersion. Last assertion is a consequence of Theorem 3.4.1.4. We can thus reduce ourselves to the case X = Spf(A), where A 2 CAlg ad k is a t-complete topological of finite presentation derived k -algebra. We wish to prove that Spf(A) rig is a derived k-affinoid space. Let C denote the full subcategory of dfDM k spanned by those affine derived k -adic formal Deligne-Mumford stacks Spf(A) such that Spf(A) rig is equivalent to a derived k-affinoid space. We have:

⇡ i O rig Y is a coherent sheaf over ⇡ 0 O rig Y . The latter is a retract of ⇡ i O rig X , which is a coherent sheaf over ⇡ 0 O rig X .
In this way, it follows that

⇡ i O rig Y is coherent over ⇡ 0 O rig X . As ⇡ 0 O rig Y is a retract of ⇡ 0 O rig X we deduce that ⇡ i O rig Y is coherent over ⇡ 0 O rig Y , as desired.
Let now X 2 dfDM taft k be an affine object. Write X ' Spf(A) for some adic derived k -algebra A 2 CAlg ad k topologically almost of finite presentation. We wish to prove that X 2 C. Theorem 3.4.1.4 guarantees that t 0 X rig is a k-analytic space. We are thus reduced to show that

⇡ i O rig X is a coherent sheaf over ⇡ 0 O rig X .
For every n 0 the algebra ⌧ n (A) is a compact object in the 1-category CAlg ad k n of n-truncated derived adic k -algebras. We can thus find a finite diagram of free simplicial k -algebras

g : I ! CAlg k , such that ⌧ n A is a retract of ⌧ n (B), where B := colim I (g) t 2 CAlg ad k ,
where (g) t denotes the t-completion of the diagram g : I ! CAlg k . As the t-completion functor commutes with finite colimits it follows that B ' B t , and in particular B is t-complete. As C is closed under finite limits and objects in the pregeometry T ad (k ), we conclude that Spf(B) 2 C. In particular Spf (⌧ n B) 2 C. As C is moreover closed under retracts, it follows that Spf (⌧ n A) 2 C as well. It follows, that for each

0  i  n, ⇡ n O rig X is coherent over ⇡ 0 O rig X .
Repeating the argument for every n 0 we conclude.

Rigidification of structures

Construction 3.4.2.1. Let X = (X, O) 2 Top R (T an (k)) be a T an (k)-structured 1-topos. Suppose further that there exists X = (Z, O 0 ) 2 Top R (T ad (k )) such that we have an equivalence X rig ' X in Top R (T an (k)).
Precomposition along the transformation of pregeometries

( ) rig : T ad (k ) ! T an (k)
induces a functor at the level of 1-categories of structures

( ) + : AnRing k (X) /O ! fCAlg k (X) /O +
given on objects by the formula

A 2 AnRing k (X) /O 7 ! A + := A ( ) rig 2 fCAlg k (X) /O + .
The functor of presentable 1-categories ( ) + : AnRing k (X) /O ! fCAlg k (X) /O + preserves limits and filtered colimits. Thanks to the Adjoint functor theorem it follows that there exists a left adjoint

( ) rig, : fCAlg k (X) /O + ! AnRing k (X) /O . (3.4.2.1)
The counit of the adjunction ( ) + , ( ) rig : Top

R (T ad (k )) ! Top R (T an (k)
) produces a well defined, up to contractible indeterminacy, morphism

f : X + = (X, O + ) ! (Z, O 0 ) = X (3.4.2.2) in the 1-category Top R (T ad (k )). Let (f 1 , f ⇤ ) ! X ! Z denote
the underlying geometric morphism associated to f . Then f 1 : Z ! X induces a well defined functor

f 1 : fCAlg k (Z) /O0 ! fCAlg k (X) /f 1 O0 . (3.4.2.3)
Moreover, the morphism (3.4.2.2) induces a morphism at the level of structures

✓ : f 1 O 0 ! O + ,
which induces a well defined functor at the level of 1-categories of structures

✓ : fCAlg k (X) /f 1 O0 ! fCAlg k (X) /O + (3.4.2.4)
given on objects by the formula

A ! f 1 O 0 2 fCAlg k (X) /f 1 O0 7 ! A ! O + 2 fCAlg k (X) /O + .
Therefore the composite ( ) rig := ( ) rig, ✓ f 1 induces a functor

( ) rig : fCAlg k (Z) /O0 ! AnRing k (X) /O
which we refer to as the rigidification functor at the level of structures.

T disc (k ) T disc (k) T ad (k ) T an (k) ⌦ k k ( ) t ( ) an ( ) rig (3.4.2.5)
Notice that (3.4.2.5) is not commutative. The lower composite sends

A 1 k 2 T disc 7 ! A 1 k 2 T an (k)
whereas the top composite sends

A 1 k 2 T disc 7 ! B 1 k 2 T an (k), where B 1 k 2 T an (k) denotes the closed unit disk. Let A 2 fCAlg k (Z) /O0
, the counit of the adjunction (( ) + , ( ) rig ) induces a natural morphism at the level of T ad (k )-structures on X

✓ A : sp 1 A ! A rig,+ := A rig + .
Applying the underlying algebra functor ( ) alg : fCAlg k (X) /O + ! CAlg k (X) /O +,alg to the morphism ✓ A we obtain a morphism

✓ alg A : (sp 1 A) alg ! A rig,+,alg (3.4.2.6)
in the 1-category CAlg k (X) /O +,alg . As A rig,+,alg lives naturally over the non-archimedean field k we obtain by adjunction a morphism

✓ alg A : (sp 1 A) alg ⌦ k k ! A rig,+,alg := A rig,+ alg , in the 1-category CAlg k (X) O +,alg . We can identify A rig,+,alg ' A(B 1 k ). There is a natural inclusion of k-analytic spaces B 1 k ! A 1 k .
We obtain thus a canonical morphism

A(B 1 k ) ! A(A 1 k ), (3.4
✓ A : (sp 1 A) alg ⌦ k k ! A rig (A 1 k ) (3.4.2.8)
in the 1-category CAlg(X) /O(A 1 k ) . We will take as (a probably confusing) convention to denote precomposition with ( ) an in (3.4.2.5) by

( ) alg : AnRing k (X) /O ! CAlg k (X) /O alg .
In this case, we might as well write (3.4.2.8) as

✓ A : (sp 1 A) alg ⌦ k k ! A rig,alg := A rig alg .
Proposition 3.4.2.5. Let X = (X, O) 2 dAn and suppose there exists X = (Z, O 0 ) 2 dfDM k such that X rig ' X, in the 1-category dAn. Then for every A 2 fCAlg k (Z) /O0 the natural morphism

✓ A : (sp 1 A) alg ⌦ k k ! A rig,alg introduced in Theorem 3.4.2.4 is an equivalence in the 1-category CAlg k (X) /O alg .
Proof. Both the underlying 1-topoi of X and X have enough points, as these are hypercomplete and 1-localic. Therefore, thanks to [Por15c, Theorem 1.12], we are reduced to check the statement of the proposition on stalks, (notice that given a geometric point x ⇤ : S ! X the composite sp ⇤ x ⇤ : S ! Z is also a geometric point).

By doing so, we might assume from the start that X = S = Z. Both composites ( ) alg ( ) rig and ( ) alg sp 1 ⌦ k k commute with sifted colimits. The proof of Theorem 3.3.1.11 implies that the 1-category fCAlg k (S) /O0 is generated under sifted colimits by the family { (k [T 1 , . . . , T m ])} m , where the T i 's sit in homological degree 0. It thus suffices to show that

✓ A : (sp 1 A) alg ⌦ k k ! A rig,alg is an equivalence whenever A ' k [T 1 , . . . , T m ]
. But in this case, we have natural equivalences

(sp 1 (k [T 1 , . . . , T m ]) alg ⌦ k k ' khT 1 , . . . , T m i,
and as

(k [T 1 , . . . , T m ]) can be identified with (a germ) of A m k 2 Top R (T ad (k )) it follows that (k [T 1 , . . . , T m ]) rig,alg ' khT 1 , . . . , T m i, in the 1-category CAlg k (X) /O alg .
The result now follows.

Rigidification of modules

Definition 3.4.3.1. Let X = (X, O) 2 dAn be a derived k-analytic space. Its 1-category of modules is defined as

Mod O := Sp Ab AnRing k (X) /O
One has the following result:

Mod O ' Mod O alg . Lemma 3.4.3.3. Let X = (X, O) 2 Top R (T an (k)) and X = (Z, O 0 ) 2 Top R (T ad (k )) such that X rig ' X in the 1-category Top R (T an (k)).
The rigidification functor ( ) rig : fCAlg k (Z) /O0 ! AnRing k (X) /O induces a well defined functor, up to contractible indeterminacy, ( ) rig : Mod O0 ! Mod O which we shall refer to as the rigidification of modules functor.

Proof. It suffices to show that the functor ( ) rig : fCAlg k (Z) /O0 ! AnRing k (X) /O commutes with finite limits. Thanks to Theorem 3.4.2.5 the composite functor ( ) alg ( ) rig agrees with localization at t and therefore it commutes with finite limits. As ( ) alg is a conservative right adjoint it follows that ( ) rig : fCAlg k (Z) /O0 ! AnRing k (X) /O commutes with finite limits as well, and the proof is finished.

We have a natural projection functor ⌦

Map Mod O ⇣ L an O/A , M ⌘ ' Der an A (O, M) . We can describe explicitly L an O/A ' ⌃ 1 an (O ⌦ A O) 2 Mod O . Lemma 3.4.3.4. Let X = (X, O) 2 Top R (T an (k)) and X = (Z, O 0 ) 2 Top R (T ad (k )) such that X rig ' X in the 1-category Top R (T an (k)). Then the diagram Mod O0 Mod O fCAlg k (Z) /O0 AnRing k (X) /O ( ) rig ⌃ 1 ad ( ) rig ⌃ 1
an is commutative up to coherent homotopy.

Proof. It suffices to prove that the corresponding diagram of right adjoints

Mod O0 Mod O fCAlg k (Z) /O0 AnRing k (X) /O ⌦ 1 ad ( ) + ⌦ 1 ad ( ) +
is commutative. But this is immediate from the definitions and the result follows.

Corollary 3.4.3.5. We have a natural equivalence

L ad O/A rig ' L an O rig /A rig in the 1-category Mod O .
Proof. It is an immediate consequence of Theorem 3.4.3.4 above.

Definition 3.4.3.6. Let M 2 Coh + (X). We say that M admits a formal model if there exists an O 0 -module

M 0 2 Coh + (O 0 ) such that M rig 0 ' M 2 Mod O . Proposition 3.4.3.7. Let X = (Z, O 0 ) 2 dfDM

Main results

In this §we our two main results. The first one concerns the existence of formal models for quasi-paracompact and quasi-separated derived k-analytic spaces. The second is a direct generalization of Raynaud's localization theorem. Definition 3.4.4.1. Let A 2 CAlg ad k . We say that A is an admissible adic derived k -algebra if A is topologically almost of finite presentation and t-complete and moreover, for every i 0, the homotopy sheaf ⇡ i (A) is t-torsion free. We denote CAlg adm Definition 3.4.4.3. Let X = (X, O) be a derived k-analytic space. We say that X is quasi-paracompact and quasi-separated if the 0-th truncation t 0 (X) is equivalent to a quasi-paracompact and quasi-separated ordinary k-analytic space. Definition 3.4.4.4. Let X = (X, O) be a derived k-analytic space. We say that X admits a formal model if there exists X 2 dfDM k such that

X ' X rig ,
in the 1-category dAn.

Thanks to [Bos05, Theorem 3, page 204] it follows that if t 0 (X) is quasi-paracompact and quasi-separated then it admits a classical formal model. We generalize this result to the derived setting: Theorem 3.4.4.5. Let X = (X, O) be a quasi-paracompact and quasi-separated derived k-analytic space. Then X admits a derived formal model X = (Z, O 0 ) 2 dfSch.

Proof. Let X 0 := t 0 (X) denote the 0-truncation of X. Thanks to [Bos05, Theorem 3, page 204] it follows that X 0 admits a formal model X 0 2 fSch k such that X 0 is admissible (i.e. it can be Zariski locally covered by affine formal spectrums of admissible k -algebras). We inductively construct a sequence of derived admissible k -adic schemes

X 0 ! X 1 ! X 2 ! . . . ,
such that we have equivalences

(X n ) rig ' t n (X),
for each n 0.The case n = 0 being already dealt it suffices to treat the inductive step. Suppose X n = (Z, O 0,n has already been constructed, for n 0. As X is a derived k-analytic space, for each n 0 the homotopy sheaf

⇡ n (O) is a coherent module over ⇡ 0 (O).
Thanks to [PY17a, Corollary 5.42] there exists an analytic derivation d :

L an t n X ! ⇡ n+1 (O)[n + 2] together with a pullback diagram ⌧ n+1 O ⌧ n O ⌧ n O ⌧ n O ⇡ n+1 O)[n + 2], d0 (3.4.4.1)
in the 1-category AnRing k (X) /⌧ n O . Here d 0 denotes the trivial analytic derivation. Theorem .1.2.1 and its proof imply that we can find a formal model for d in the stable

1-category Coh + (X n ) : L ad Xn ! M n+1 [n + 2],
where M n+1 2 Coh + (X n ) ~is of no t-torsion and we have that

M rig n+1 ' ⇡ n+1 (O) in the 1-category Coh + (⌧ n X). We define O n+1 2 fCAlg k (Z) /O0,n as the pullback of the diagram O 0,n+1 O 0,n O 0,n O 0,n M n+1 [n + 2] d0 (3.4.4.2) in fCAlg k (Z) /O0,n . Define X n+1 := (Z, O 0,n+1
↵ n+1 : (O 0,n+1 ) rig ! ⌧ n+1 O in the 1-category AnRing k (X) /⌧ n+1 O .
We have thus a canonical morphism

✓ n+1 : t n+1 (X) ! X rig n+1 .
We claim that ✓ n+1 is an equivalence in the 1-category Top R (T an (k)). It suffices to show that ↵ n+1 is an equivalence of structures. Thanks to Theorem 3.4.2.5 we have an equivalence

⇣ O rig 0,n+1 ⌘ alg ' sp 1 O 0,n+1 alg ⌦ k k.
By the inductive hypothesis together with the pullback diagrams (3.4.4.2) and it follows that

(sp 1 O 0,n+1 ) alg ⌦ k k ' (⌧ n+1 O) alg
is an equivalence. By conservativity of ( ) alg it follows that

↵ n+1 : O rig 0,n+1 ' ⌧ n+1 O
in the 1-category AnRing k (X) /⌧ n+1 O . We conclude that

✓ n+1 : X rig n+1 ' t n+1 X is an equivalence in Top R (T an (k))
. We define

X := colim n 0 X n .
We claim that X is again an admissible derived k -adic Deligne-Mumford stack: the question being local on X reduce ourselves to the case X = Spf A and X n ' Spf A n , for A, A n 2 CAlg adm k , for each n 0. By construction, T n 1 A n ' A n 1 for each n 1. We have moreover an identification

X ' Spf (lim n 0 A n ) .
As A n is admissible we conclude that lim n 0 A n is also admissible. We have thus proved that X is an admissible derived k -adic Deligne-Mumford stack.

We are finished if we prove that X rig ' X. We have a sequence of equivalences

t n (X rig ) ' (t n X) rig ' X rig n ' t n (X)
by convergence of derived k-analytic stacks, see [PY17a,[START_REF] Clark | Fundamental Groups in Characteristic p[END_REF]. Assembling these equivalences together produces a map f : X rig ! X in the 1-category dAn. The underlying morphism of 1-topoi is an equivalence. The morphism f induces equivalences, for each i 0,

⇡ i (O rig 0 ) ' ⇡ i (O)
, where O 0 := lim n 0 O 0,n . By hypercompletion of the X it follows that O rig 0 ' O. Thus proving that f is an equivalence, finishing the proof.

We now deal with our main result. We start with a useful lemma: Lemma 3.4.4.6. Let F : C ! D be a functor between 1-categories. Suppose that for any D 2 D the following assertions are satisfied: Proof. Let E be an 1-category. We have to prove that precomposition along F induces a fully faithful embedding of 1-categories

(i) The 1-category C /D := C ⇥ D D /D is contractible; (ii) let C 0 /D
F ⇤ : Fun (D, E) ! Fun (C, E) whose essential image consists of those functors G : C ! E which send morphisms in S to equivalences in D.
Given any functor G : D ! E, the composite G F : C ! E sends each morphism in S to an equivalence E as F does (in D). Thanks to the colimit formula for left Kan extensions together with conditions (i) and (ii) in the statement of the Lemma, we conclude that given a functor G : C ! E such that any morphism in S is sent to an equivalence, its left Kan extension F ! (G) 2 Fun (D, E) exists and we have natural. equivalence F ! F ⇤ ' id and F ⇤ F ! ' id. The result now follows from the fact that F ! is an inverse to F ⇤ when restricted to the full subcategory of Fun (C, E) spanned by those functors sending every morphism in S to an equivalence in E. ] is typically a genuine 1-category.

Definition 3.4.4.8. Let dAn 0 ✓ dAn denote the full subcategory of dAn spanned by those quasi-paracompact and quasi-separated derived k-analytic spaces X 2 dAn. Definition 3.4.4.9. Let f : X ! Y be a morphism between derived k -adic schemes. We say that f is generically strong if for each i > 0 the induced morphism Proof. The verification of the assumptions of Theorem 3.4.4.6 are made simultaneously: Let X 2 dAn 0 , C X := dfSch adm X/ and p 0 : C X ! dfSch, p 1 : C X ! dAn 0 denote the canonical projections. We will show that for every finite space K and every functor f :

⇡ i (f ⇤ O Y ) rig ! ⇡ i (O X ) rig is an equivalence in the 1-category Coh + (X rig ).
K ! C X we can extend f to a (cone) functor f C : K C ! C X in such a way that f C
(1) is a formal model for X 2 dAn 0 , where 1 2 K C denotes the cone point. This will imply that C X is a cofiltered 1-category, hence of contractible homotopy type and moreover the inclusion of the full subcategory of formal models for X is final in C X .

Let us first sketch the rough idea of proof: By induction on Postnikov towers we allow ourselves to lift commutative diagrams of derived k-analytic spaces to the formal level. This is done, by reducing questions of lifting of T ad (k )-structures on certain 1-topoi to lifting questions at the level of 1-categories of coherent modules, using the universal property of the adic cotangent complex. The corresponding questions for coherent modules can be dealt using the refined results in Appendix A. The main technical difficulty is thus keeping track of higher coherences for commutative diagrams when passing from the analytic 1-category to the k -adic one.

We will construct a sequence {(X n , t n X ! X rig ) 2 C t n X } n2N such that X n := (X n , O Xn ) 2 dfSch adm satisfies the following conditions:

(i) For each n 0, X n is n-truncated.

(ii) For each n 0, we have an equivalence

X n rig ' t n X.
(iii) For each n 0, we have a canonical morphism

X n ! t n+1 X n
in the 1-category dfSch adm which is moreover an equivalence. This implies, in particular, that the underlying 1-topoi X n 2 Top R are all equivalent, for n 0.

(iv) For each n 0, there is a functor

f C n 2 Fun K C , C tnX whose restriction (f C n ) |K is naturally equivalent to t n f in the 1-category Fun K, C t n X and such that p 0 (f C n (1)) ' X n .
Assume that we have constructed such a sequence {(X n , t n X ! X rig ) 2 C t n X } n2N satisfying conditions (i) through (iv). Define X := colim n 0 X n and notice that in such case the morphisms t n X ! X rig n assemble to induce a morphism X ! X rig , in the 1-category dAn. Moreover, by the universal property of filtered limits the diagrams f C n 2 Fun K C , C t n X assemble thus producing a well defined (up to contractible indeterminacy) extension

f C 2 Fun (K C , C X ) of f : K ! C X .
As the rigidification functor is compatible with n-truncations it follows that the functor f C obtained in this way implies that p 1 f C

(1) 2 dAn 0 X/ corresponding to the morphism

X ! X rig ,
in the 1-category dAn, is an equivalence. This finishes the proof of the claim. Therefore, we are reduced to prove the existence of a sequence {(X n , t n X ! X rig n ) 2 C t n X } n2N satisfying conditions (i) through (iv) above.

Step 1

(Case n = 0) Let X 0 := t 0 X 2
An denote the underlying ordinary k-analytic space to X. By the universal property of n-truncation we can assume without loss of generality that for each vertex Step 2 (Inductive assumptions) Suppose now, that for n

x 2 K the component Y x , x : X 0 ! Y rig x := f (x) 2 C X0 is actually discrete, i.e. Y X
0 we have constructed a diagram f C n 2 Fun K C , C t n X satisfying conditions (i) through (iv) above. Denote by ↵ n,x : X n ! Y n,x the morphism associated to 1 ! x in K C , where Y n,x := t n Y x = Y n,x , O n,x . The functor f C n 2 Fun K C , C t n X corresponds to the following given: (i) A diagram f C n,⇤ : K C ! Top R such that f (1) ' X n and for each x 2 K a morphism ↵ n,x,⇤ : X n ! Y n,x in Top R . We remark that this data is constant for 0  m  n. (ii) A diagram f C, 1 n : K C,op ! fCAlg k (X n ) /O Xn such that f C, 1 (1) ' id O Xn and f C, 1 (x) corresponds to a morphism h n,x : ↵ 1 n,x O Yn,x ! O Xn in the 1-category fCAlg k (X n ) /O Xn .
A similar analysis for the diagram t n+1 f : K ! C t n+1 X together with the Postnikov decomposition imply that we have a functor f 1 n+1 :

K op ⇥ 1 2 ! fCAlg k (X n ) /O Xn such that for each x 2 K the induced morphism f 1 n+1,x : 1 2 ! fCAlg k (X n ) /O Xn corresponds to a pullback diagram of the form ⌧ n+1 ↵ 1 x O Yx ⌧ n ↵ 1 x O Yx ⌧ n ↵ 1 x O Yx ⌧ n ↵ 1 x O Yx ↵ 1 x ⇡ n+1 O Yx [n + 2] dn,x d 0 n,x (3.4.4.3)
in the 1-category fCAlg k (X n ) /O Xn , where d n,x denotes a suitable k -adic derivation and d 0 n,x the trivial adic derivation.

Step 3 (Functoriality of the construction fCAlg k (X) O//O ) Consider the functor I : fCAlg k (X n ) /O Xn ! Cat 1 given on objects by the formula

O ! O Xn 2 fCAlg k (X n ) /O Xn 7 ! fCAlg k (X n ) O//O 2 Cat 1
O : fCAlg k (X n ) O Xn //O Xn ! fCAlg k (X n ) O//O , which admits a left ad- joint f O : fCAlg k (X n ) O//O ! fCAlg k (X n ) O Xn //O Xn , obtained via base change along O ! O Xn .
Therefore, applying the unstraightening construction, we obtain a well defined functor

G : fCAlg k (X n ) O Xn //O Xn ⇥ fCAlg k (X n ) /O Xn ! D over fCAlg k (X n ) /O Xn , whose fiber at (O ! O Xn ) 2 fCAlg k (X n ) /O Xn coincides with g O introduced above. Thanks to the (dual) discussion proceding [PY16c, Corollary 8.6] it follows that G admits a left adjoint F : D ! fCAlg k (X n ) O Xn //O Xn ⇥ fCAlg k (X n ) /O Xn .
Step 4

K op ! fCAlg k (X n ) O Xn //O Xn , respectively. Notice that 0 : K op ! fCAlg k (X n ) O Xn //O Xn
is given on objects by the formula

x 2 K op 7 ! ✓ O Xn ! O Xn ↵ ⇤ x ⇡ n+1 O Yx [n + 2] d 0 n,x ! O Xn ◆ 2 fCAlg k (X n ) O Xn //O Xn
and similarly for :

K op ! fCAlg k (X n ) O Xn //O Xn we have x 2 K op 7 ! ✓ O Xn ! O Xn ↵ ⇤ x ⇡ n+1 O Yx [n + 2] dn,x ! O Xn ◆ 2 fCAlg k (X n ) O Xn //O Xn .
By construction, both functor 0 and factor through the full subcategory

fCAlg der k (X n ) O Xn //O Xn ✓ fCAlg k (X n ) O Xn //O Xn spanned by those objects O Xn ! A ! O Xn which correspond to k -adic derivations.
Step 5

(Reduction of the above diagrams to diagrams of modules) The universal property of the k -adic cotangent complex implies that we have an equivalence of 1-categories

( ) der : fCAlg der k (X n ) O Xn //O Xn ' Mod O Xn L ad O Xn / .
Therefore, the functors 0 and as above correspond, under the equivalence ( ) der , to functors 0 , :

K op ! Mod O Xn L ad O Xn
given on objects by the formulas

x 2 K op 7 ! d 0 n,x : L ad O Xn ! ↵ ⇤ x ⇡ n+1 O Yx [n + 2] 2 Mod O Xn L ad O Xn / and x 2 K op 7 ! d n,x : L ad O Xn ! ↵ ⇤ x ⇡ n+1 O Yx [n + 2] 2 Mod O Xn L ad O Xn / ,
respectively. Thanks to the proofs of both [PY17a, Lemma 5.35 and Corollary 5.38] the k -adic cotangent complex L ad O Xn is coherent and connective. Therefore the functors 0 , :

K op ! Mod O Xn L ad O Xn /
factor through the full

subcategory Coh + (O Xn ) L ad O Xn / ✓ Mod O Xn L ad O Xn / .
Step 6

(Rigidification of the corresponding diagrams of modules) Consider now the composites

rig 0 := ( ) rig 0 , rig := ( ) rig : K op ! Coh + (O rig Xn ) L an X rig n / .
The same reasoning as above applied to the rigidification of the diagram

t n+1 f : K ! C t n+1 X produces extensions e rig 0 , e rig : K C,op ! Coh + (O rig Xn ) L an O rig Xn
of ( ) rig 0 and of ( ) rig , respectively, satisfying:

(i) We have equivalences f 0 |K op ' ( ) rig 0 , e |K op ' ( ) rig in the 1-category Fun K op , Coh + (O rig Xn ) L an X rig n / .
(ii) We have moreover equivalences

f 0 rig (1) ' ✓ d 0 : L an O rig Xn ! ⇡ n+1 O X [n + 2]

◆

, and e rig (1) '

✓ d : L an O rig Xn ! ⇡ n+1 O X [n + 2] ◆ in the 1-category Coh + (t n (X)) L an t n X / .
Where the derivations d 0 and d considered above are induced by the pullback diagram

⌧ n+1 O X ⌧ n O X ⌧ n O X ⌧ n O X ⇡ n+1 O X [n + 2] d d0 (3.4.4.4) in the 1-category AnRing k (X) /O X .
Step 7

(Lifting of f 0 rig and e rig to diagrams in Coh + (X n ).) Thanks to Theorem .1.2.1 and its proof, we can lift both diagrams e rig 0 and e rig to (formal model) diagrams 0 , :

K op ! Coh + (O Xn ) L ad O Xn /
, respectively. We have moreover equivalences

0|K op ' 0 , |K op ' . and 0 (1) ' ⇣ 0 : L ad O Xn ! N [n + 2] ⌘ (3.4.4.5) (1) ' ⇣ : L ad O Xn ! N [n + 2] ⌘ (3.4.4.6)
where

N 2 Coh + (O Xn ) denotes a t-torsion free formal model of ⇡ n+1 (O Xn ), concentrated in degree 0. The choice of such N 2 Coh + (O Xn ) can be realized as follows: First choose a given formal model N 2 Coh + (O Xn ) for ⇡ n+1 (O X ).
As the rigidification functor ( ) rig is compatible with n-truncations, we can replace N with ⌧ n N and thus suppose that N is truncated to begin with. We can kill the t-torsion on N by multiplying it by a sufficiently large power of t, i.e. consider t m N for m > 0 sufficiently large such that t m N is t-torsion free. The conclusion now follows thanks to the fact that the canonical map t m N ! N induces an equivalence t m N rig ' N rig .

Step 8

(
C,op ! fCAlg k (X n ) O Xn //O Xn ⇥ fCAlg k (X n ) /O
Xn whose projection along the first component agrees with 0 and , respectively, and whose projection along the second component agrees with the composition F f C, 1 . By adjunction, we obtain thus diagrams D 0 , D :

K C,op ! D inducing D 0 0 , D 0 : K C,op ⇥ 2 ! fCAlg k (X n ) O Xn / given on vertices x 2 K by the formula x 2 K op 7 ! ✓ ⌧ n ↵ 1 x O Yx d0,n ! ⌧ n ↵ 1 x O Yx ⇡ n+1 O Yx [n + 2] ! ⌧ n ↵ 1 x O Yx ◆ 2 fCAlg k (X n ) O Xn / x 2 K op 7 ! ⇣ ⌧ n ↵ 1 x O Yx dn ! ⌧ n ↵ 1 x O Yx ⇡ n+1 O Yx [n + 2] ! ⌧ n ↵ 1 x O Yx ⌘ 2 fCAlg k (X n ) O Xn / ,
respectively. Moreover, their value at 1 correspond to

O Xn d0 ! O Xn N [n + 2] ! O Xn , O Xn d ! O Xn N [n + 2] ! O Xn , respectively.
Step 9

(Obtaining an extension f C n+1 of the diagram t n+1 f ) By taking fiber products along over each {x} ⇥ ⇤ 2 2 we obtain thus a diagram f C n+1 : K C,op ! fCAlg k (X n ) O Xn whose value on each x 2 K agrees with

f C n+1 (x) ' ⌧ n+1 ↵ 1 x O Yx .
More precisely, we have a canonical equivalence

f C n+1 |K ' ⌧ n+1 f 1 . Moreover, for f C n+1 (1) ' O n+1 2 fCAlg k (X n ) such that O rig n+1 ' ⌧ n+1 O X , in the 1-category AnRing k (X). Let X n+1 := (X n , O n+1 ).
We obtain thus a well defined functor

f C n+1 : K C ! dfSch adm
whose rigidification coincides with

⌧ n+1 f : K ! dAn 0 t n+1 X/ .
Assembling these diagrams together we obtain a functor f C n+1 : K C ! C X satisfying requirements (i) through (iv) above, which concludes the proof.

The proof of Theorem 6.2.3.15 also implies: Corollary 3.4.4.12. Let f : X ! Y be a morphism between quasi-paracompact and quasi-separated derived k-analytic spaces. Then f admits a formal model, i.e. there exists a morphism f : X ! Y in dfSch adm such that f rig ' f in the 1-category dAn. Proof. The result is a direct application of the proof of Theorem 3.4.4.10 when X 2 dAfd.

.

Verdier quotients and Lemma on Coh +

The results in this section were proved in a joint work with M. Porta on the representability of the derived Hilbert stack, many of the statements and proofs are due to him.

.

Verdier Quotients

In this §we let X be a quasi-compact and quasi-separated scheme and Z denote the formal completion of X along the (t)-locus. Consider also Z rig 2 An its rigidification. We have a rigidification functor at the level of the derived 1-categories of almost perfect complexes

( ) rig : Coh + Z ! Coh + Z rig .
Notation . Proposition .1.1.4. Let X be a quasi-compact quasi-separated derived scheme almost of finite type over k . We denote Z its formal (t)-completion and Z rig 2 dAfd its rigidification. Then there exists a cofiber sequence

K(Z) ,! Coh + (Z) ! Coh + Z rig (.1.1.1)
in the 1-category Cat Ex 1 . Moreover, the functors in (.1.1.1) are t-exact. In particular, the rigidification functor

( ) rig : Coh + (Z) ! Coh + Z rig
exhibits Coh + Z rig as a (t-exact) Verdier quotient of Coh + (Z).

Proof. Let K(Z) denote the full subcategory of Coh + (Z) spanned by t-torsion almost perfect modules on Z.

Recall that M 2 Coh + (Z) is of t-torsion if ⇡ ⇤ (M ) is of t-torsion. Consider the (quasi-compact) étale site X ét of X. We define a functor Coh + ( )/K( ) : X ét ! Cat Ex 1
given on objects by the formula

U ! X quasi-compact and étale 7 ! Coh + (U t )/K(U t ) 2 Cat Ex 1
where U t denotes the formal completion of U along the (t)-locus. Thanks to [HPV16a, Theorem 7.3] this defines a uniquely, up to contractible indeterminacy, defined Cat Ex 1 -valued sheaf for the étale topology. We will also need the following ingredient: define a functor given on objects by the formula

Coh + rig : X ét ! Cat Ex
U ! X quasi-compact and étale 7 ! Coh + (U t ) rig 2 Cat Ex 1 .
We remark that Coh + : An ! Cat Ex 1 satisfies fpqc descent for k-analytic spaces which follows by the main theorem in [Con03a] together with the usual reasoning by induction on the Postnikov towers for almost perfect modules in order to reduce the statement for Coh + to a similar statement concerning the heart Coh +,~. Moreover, the formal completion and rigidification functors are morphisms of sites. As a consequence we conclude that the assignment Coh + r ig : X ét ! Cat Ex 1 is a sheaf for the étale topology on X. The universal property of pushout induces a canonical morphism of sheaves : Coh + ( )/K( ) ! Coh + rig in the 1-category Shv ét (X, Cat Ex 1 ). We affirm that is an equivalence in Shv ét (X, Cat Ex 1 ). By descent, it suffices to prove the statement on affine objects of X ét . In such case, the result follows readily from the observation that for a derived k -algebra A 0 the 1-category

Coh + A 0 ⌦ k k 2 Cat Ex 1 is obtained from Coh + (A 0 )
by "modding out" the full subcategory spanned by t-torsion almost perfect modules. Moreover, thanks to [PY18a, Theorem 3.1] we have a canonical equivalence

Coh + (Spf A 0 ) rig ' Coh + (Spf A 0 ) rig
in the 1-category Cat Ex 1 , where (Spf A 0 ) rig 2 CAlg k denotes the derived global sections of Spf A rig 0 . On the other hand (Spf A 0 ) rig ' A 0 ⌦ k k and the result follows.

.

Existence of formal models for modules

In this §we prove some results concerning the existence of formal models with respect to the functor ( ) rig : Coh + (X) ! Coh + (X) which prove to be fundamental in the proof of Theorem 3.4.4.10. I am thankful to Mauro Porta as the results in this §were proved in a joint work.

Proposition .1.2.1. Let X 2 dAn be a derived k-analytic stack admitting a formal model X 2 dfSch, i.e.

(X) rig ' X in dAn. Let F 2 Coh + (X) be concentrated in finitely many cohomological degrees. Then F admits a formal model, i.e. there exists G 2 Coh + (X) such that G rig ' F in Coh + (X). Moreover, the 1-category of those formal models for F is a filtered 1-category.

Proof. Let F 2 Coh + (X), be as in the stament of the Theorem .1.2.1. Assume moreover that F is connective, i.e. its non-zero cohomology lives in non-positive degrees. Notice that, by definition of ind-completion, F 2

Ind Coh + (X) is a compact object. Let : Ind(Coh + (X)) ! Ind(Coh + (X)
) denote a fully faithful right adjoint to ( ) rig . It follows from the construction of Ind-completion that we have a canonical equivalence

(F) ' colim G2Coh + (X) / (F) G, (.1.2.1)
in Ind Coh + (X) , where, by construction, the limit indexing 1-category appearing on the right hand side of (.1.2.1) is filtered. As is a fully faithful functor, the counit of the adjunction ( ) rig , is an equivalence. Our argument now follows by an inductive reasoning using the Postnikov tower as we now detail: Suppose first that F 2 Coh + (X) has cohomology concentrated in degree 0, then it is well known that F admits a formal model e F 2 Coh +,~( X), which we can moreover choose to be of no t-torsion. Moreover, we can choose e F in such a way that we have a monomorphism e F ,! F in the heart Coh +,~( X)), whose rigidification becomes an equivalence, in the (heart of) Ind Coh + (X) . We are then dealt with the base of our inductive reasoning. Suppose now that F lives in cohomological degrees [ n, 0], by the inductive hypothesis

T n 1 F 2 Coh + (X) admits a formal model T n 1 F 2 Coh + (X), which lives in cohomological degrees [ n + 1, 0]
and is moreover of no t-torsion and we have a map T n 1 F ! T n 1 F in the 1-category Ind Coh + (X) , whose rigidification becomes an equivalence. We have a fiber sequence

F T n 1 F ⇡ n (F) [n + 1],
in the 1-category Coh + (X). By applying the exact functor we also obtain a fiber sequence in the 1-category

Coh + (X). As T n 1 F 2 Ind Coh + (X) is a compact object, the composite T n 1 F ! T n 1 F ! ⇡ n (F) [n + 1] factors through G[n + 1], for an almost perfect complex G 2 Coh + (X) ~, such that G rig ' ⇡ n (F)
, which by the base step, we can choose to be of no p-torsion and admitting a monomorphism G ! ⇡ n (F) in the heart of the 1-category Ind Coh + (X) .

Using the fact that is a right adjoint and the counit is an equivalence, the rigidification of the constructed map

T n 1 F ! G[n + 1] is equivalent to T n 1 F ! ⇡ n (F)[n + 1]. Therefore e F := fib ⇣ T n 1 F ! G[n + 1]
⌘ is a formal model for F, which lives in cohomological degrees

[ n, 0], of no t-torsion and admitting a map e F ! F in the 1-category Ind Coh + (X) , which become an equivalence after rigidification. The first part of Theorem .1.2.1 now follows.

We are now left to prove that the full subcategory C F of the filtered 1-category Coh + (X) /F spanned by those objects ⇣ e F, : e F rig ! F ⌘ such that is an equivalence, is also filtered.

By construction, the 1-category Coh + (X) /F is filtered. In order to prove that C F is filtered, it suffices to show that every G, : G rig ! F 2 Coh + (X) /F admits a morphism to an object in C F . We first treat the case where F 2 Coh + (X) lies in the heart so then we can write F ' colim i2I G i in Ind Coh + (X) ~, where I is filtered. Moreover, we can assume that the G i 2 Coh + (X) ~are (of no t-torsion) and for each i 2 I they admit monomorphisms G i ! F such that after rigidification one has

G rig i ' F in Ind Coh + (X) ~. The structural morphism : G rig ! F corresponds by adjunction to a morphism G ! (F) ' colim i2I (G i ). By compactness of G 2 Coh +
(X) if follows that the later factors through one of the G i . To summarize, we have obtained a morphism G ! G i which induces a morphism in Coh + (X) /F whose source corresponds to G, : G rig ! F and the target is an object lying in C F , as desired.

Suppose now that F 2 Coh + (X) is connective whose non-zero cohomology lives in degress

[ n, 0]. Given G, : G rig ! F 2 Coh + (X) /F we know by induction that G, G rig ! F ! T n 1 F 2 Coh + (X) /T n 1 F admits a factorization through one object ⇣ T n 1 F, Tn 1 F rig ! T n 1 F ⌘ 2 Coh + (X) /T n 1 F ,
as before. We have a commutative diagram

⌧ n G T n 1 G ⇡ n (G)[n + 1] F T n 1 F ⇡ n (F)[n + 1],
where the horizontal maps form fiber sequences in the 1-category IndCoh + (X). Moreover, there exists a sufficiently large formal model H n 2 Coh + (X) ~for ⇡ n (F), without t-torsion together with a monomorphism H n ! ⇡ n (F) in Ind Coh + (X) ~such that both the composites

T n 1 G ! T n 1 F ! T n 1 F ! ⇡ n (F)[n + 1] and T n 1 G ! ⇡ n (G)[n + 1] ! ⇡ n (F)[n + 1]
factor through H n [n+1]. Thus we have a commutative diagram of fiber sequences in the 1-category Ind Coh + (X)

⌧ n G T n 1 G ⇡ n (G)[n + 1] e F T n 1 F M n [n + 1] F T n 1 F ⇡ n (F)[n + 1]
which provides a factorization G, : for X. Then we can find a morphism f : F 0 ! G 0 in Coh + (X) such that f rig lies in the same connected component of f in the mapping space Map Coh + (X) (F, G). Proof. We will actually prove more: Fix F 0 2 Coh + (X) a formal model for F, whose existence is guaranteed by Theorem .1.2.1 then we can find a formal model G 0 2 Coh + (X) for G such that the morphism

G rig ! F ! ⇣ e F, : e F rig ! F ⌘ in the 1-category Coh + (X) /F where ⇣ e F, : e F rig ! F ⌘ 2 C F ,
f : F ! G,
in the 1-category Coh + (X) lifts to a morphism,

f : F 0 ! G 0 , in the 1-category Coh + (X). Assume thus F 0 2 Coh + (X) fixed. Given a generic G 0 2 Coh + (X), denote by Hom (F 0 , G 0 ) 2 QCoh((X)) the Hom-sheaf of (quasi-coherent) O X -modules. Notice that if G 0 2 Coh b (X) then the Hom-sheaf Hom (F 0 , G 0
) is still an object lying in the 1-category Coh + (X) By our assumption on G 2 Coh + (X), we can find a cohomogically bounded formal model G 0 2 Coh b (X) for G, and thus

Hom (F 0 , G 0 ) 2 Coh + (X). Consider the colimit, colim G 0 2C Hom (F 0 , G 0 ) ' Hom (F 0 , G (G)) (.1.2.2) ' Hom ⇣ (F 0 ) rig , G ⌘ ' G (Hom (F, G)) , (.1.2.3)
where C denotes the 1-category of (cohomological bounded) formal models for G. The first equivalence in (.1.2.2) follows from the fact that F 0 2 Coh + (X) is a compact object in Ind(Coh + (X)), thus the Hom-sheaf, with source F 0 , commutes with filtered colimits, and the second equivalence follows from adjunction. By applying the global sections functor on both sides of (.1.2.2) we obtain an equivalence of spaces (notice that being a right adjoint respects global sections)

colim G 0 2C Map(F 0 , G 0 ) ' Map Coh + (X) (F, G).
We conclude thus that there exists G 0 2 C and f : F 0 ! G 0 such that (f) rig and f lie in the same connected component of Map (X). Suppose we are given a formal model X for X together with formal models F 0 , G 0 2 Coh + (X) for F and G, respectively, where we assume moreover that G 0 2 Coh b (X). Then given an arbitrary f :

F ! G in Coh + (X) we can find f : F 0 ! G 0 in Coh + (X) lifting t n f : F ! G, for a sufficiently large n > 0.
Proof. Consider the sequence of equivalences in (.1.2.2). Then by applying the same argument as in the proof of Theorem .1.2.1 we obtain that an equivalence,

(Hom(F 0 , G 0 )) rig ' Hom(F, G),
in the 1-category Coh + (X). Therefore, by taking global sections we obtain,

Map Coh + (X) (F 0 , G 0 )[t 1 ] ' Map Coh + (X) (F, G)
, where the left hand side term denotes the colimit colim mult by t Map

Coh + (X) (F 0 , G 0 ). Therefore, by multiplying f 2 Map Coh + (X) (F, G) by a sufficiently large power of t, say t n , then t n f should lie in a connected component of Map Coh + (X) (F 0 , G 0 ), as desired.
.

Unramifiedness of T ad (k )

In this §we prove that the k -adic pregeometry T ad (k ) together with the transformation of pregeometries ( ) t : T ét (k ) ! T ad (k ) are unramified.

Definition .2.0.1. Let T be a pregeometry. We say that T is unramified if for every morphism f : X ! Y in T and every object Z 2 T, the diagram

X ⇥ Z X ⇥ Y ⇥ Z X X ⇥ Y induces a pullback diagram X X⇥Z X X⇥Y ⇥Z X X X X⇥Y in Top R
, where X X⇥Z , X X⇥Y ⇥Z , X X and X X⇥Y denote the underlying 1-topoi associated to the absolute spectrum construction, introduced in [Lur11c, §2. Proof. Let Z 2 T ad (k ) and denote X Z denote the underlying 1-topos of the corresponding absolute spectrum Spec T ad (k ) (Z). The 1-topos X Z is equivalent to the hypercompletion of the étale 1-topos on the special fiber of Z. As pullback diagrams are preserved by taking special fibers the result follows by unramifiedness of T ét (k ).

There is also a notion of relative unramifiedness: Definition .2.0.4. Let ' : T ! T 0 be a transformation of pregeometries, and let : Top

R (T 0 ) ! Top R (
T) the induced functor given on objects by the formula

(X, O) 2 Top R (T 0 ) 7 ! (X, O ') 2 Top R (T).
We say that the transformation f is unramified if the following conditions are satisfied:

(i) Both T and T 0 are unramified;

(ii) For every morphism f : X 2 Y and every object Z 2 T, we have a pullback diagram

Spec T 0 X ⇥ Z Spec T 0 X ⇥ Y ⇥ Z Spec T 0 Z Spec T 0 X ⇥ Y in the 1-category Top R (T).
Proposition .2.0.5. The transformation of pregeometries ( )

t : T ét (k ) ! T ad (k ) is unramified.
Proof. It suffices to prove condition (ii) in Theorem .2.0.4. This follows from the fact that Spec T ad (k ) ( ) is an ind-étale spectrum, thus such construction commutes with finite limits.

.

Useful Lemma

In this § we will prove a formal statement that proved to be useful in the proof of Theorem 3. (ii) G is conservative, preserves epimorphisms and sifted colimits;

Then epimorphisms in D are also effective, moreover, if {X ↵ } is a family of compact generators for C the family {F (X ↵ )} generates D under sifted colimits.

Proof. Let g : V ! Y be an epimorphism in the 1-category D. We wan to show that it is effective, that is the canonical morphism g 0 : Y 0 := | Č(g)|Y , where Y 0 denotes the geometric realization of the Cech nerve of g, is an equivalence in D. By assumption, G(g) is an epimorphism. Since G is a right adjoint, we have a canonical equivalence

G Č(g) ' Č (G(g)) .
As G commutes with sifted colimits, we see that

G(Y 0 ) ' | Č (G(g)) | ' G(Y ).
We thus conclude that Y 0 ' Y using the conservativity of G. This finishes the proof of the first assertion.

Let Y 2 D. We can find a filtered category I and a diagram T :

I ! C such that colim ↵2I T ↵ ' G(Y ) 2 C.
Consider the composition F T : I ! D. For every ↵ 2 I, we obtain a natural map

' ↵ : F (T ↵ ) ! F (G(Y )) ! Y,
where the latter morphism is induced by the counit of the adjunction (F, G). These maps ' ↵ can be arranged into a cocone from F T to Y . For each ↵, we can form the Čech nerve Č(' ↵ ). This produces a functor e T : I ⇥ op ! D, informally defined by

(↵, n) 7 ! Č(' ↵ ) n
. There is a natural cocone from e T to Y , and we claim that the induced map

: colim (↵,n)2I⇥ op e T (↵, n) ! Y is an equivalence. We remark that colim (↵,n)2I⇥ op e T (↵, n) ' colim n2 op colim ↵2I e T (↵, n).
Since G is conservative, it is enough to check that G( ) is an equivalence. Observe that, ,since G commutes with sifted colimits, we have

G ✓ colim n2 op colim ↵2I e T (↵, n) ◆ ' colim n2 op colim ↵2I G ⇣ e T (↵, n) ⌘ .
Since I is a filtered category and G is a right adjoint, we obtain:

G ✓ colim ↵2I Č(' ↵ ) n ◆ ' Č ✓ colim ↵2I G(F (T ↵ )) ! G(Y ) ◆ n .
The unit of the adjunction (F, G) provide us with maps ⌘ ↵ :

T ↵ ! G(F (T ↵ )) such that the induced composition colim ↵2I T ↵ ! colim ↵2I G(F (T ↵ )) ! G(Y )
is an equivalence. In particular, the map

colim ↵2I G(F (T ↵ )) ! G(Y )
is an effective epimorphism. In particular,

colim ↵2I G(F (T ↵ )) ! G(Y )
is an effective epimorphism. Thus"

colim (↵,n)2 op G( e T (↵, n)) ' | Č(colim ↵2I G(F (T ↵ )) ! G(Y )| ' G(Y ).
Thus, G( ) is an equivalence, and so we conclude that was an equivalence to start with.

Contents 4.1 Introduction

Let k be a non-archimedean field equipped with a non-trivial valuation of rank 1. We let k denote its ring of integers, m an ideal of definition. We furthermore assume that m is finitely generated. Given a separated k-analytic space X, we are concerned with the existence of the derived moduli space RHilb(X), which parametrizes flat families of closed subschemes of X. The truncation of RHilb(X) coincides with the classical Hilbert scheme functor, Hilb(X), which has been shown to be representable by a k-analytic space in [CG16]. On the other hand, in algebraic geometry the representability of the derived Hilbert scheme is an easy consequence of Artin-Lurie representability theorem. In this paper, we combine the analytic version of Lurie's representability obtained by T. Y. Yu and the second author in [PY17b] together with a theory of derived formal models developped by the first author in [Ant18a]. The only missing step is to establish the existence of the cotangent complex. Indeed, the techniques introduced in [PY18b] allows to prove the existence of the cotangent complex at points

x : S ! RHilb(X) corresponding to families of closed subschemes j : Z ,! S ⇥ X which are of finite presentation in the derived sense. However, not every point of RHilb(X) satisfies this condition: typically, we are concerned with families which are almost of finite presentation. The difference between the two situations is governed by the relative analytic cotangent complex Lan Z/S⇥X : Z is (almost) of finite presentation if Lan Z/S⇥X is (almost) perfect. We can explain the main difficulty as follows: if p : Z ! S denotes the projection to S, then the cotangent complex of RHilb(X) at x : S ! RHilb(X) is computed by p + (Lan Z/S⇥X ). Here, p + is a (partial) left adjoint for the functor p ⇤ , which has been introduced in the k-analytic setting in [PY18b]. However, in loc. cit. the functor p + has only been defined on perfect complexes, rather than on almost perfect complexes. From this point of view, the main contribution of this paper is to provide an extension of the construction p + to almost perfect complexes. Our construction relies heavily on the existence results for formal models of derived k-analytic spaces obtained by the first author in [Ant18a]. Along the way, we establish three results that we deem to be of independent interest, and which we briefly summarize below.

Let X be a derived formal k -scheme topologically almost of finite presentation. One of the main construction of [Ant17b, Ant18a, ?] is the generic fiber X rig , which is a derived k-analytic space. The formalism introduced in loc. cit. provides as well an exact functor

( ) rig : Coh + (X) ! Coh + (X rig ), (4.1.0.1)
where Coh + denotes the stable 1-category of almost perfect complexes on X and on X rig . When X is underived, this functor has been considered at length in [HPV16b], where in particular it has been shown to be essentially surjective, thereby extending the classical theory of formal models for coherent sheaves on k-analytic spaces. In this paper we extend this result to the case where X is derived, which is a key technical step in our construction of the plus pushforward. In order to do so, we will establish the following descent statement, which is an extension of [HPV16b, Theorem 7.3]:

Theorem 3. The functor Coh + loc : dAn k ! Cat st 1 , which associates to every derived formal derived scheme

X 2 dfDM 7 ! Coh + (X rig ) 2 Cat st 1 ,
satisfies Zariski hyper-descent.

We refer the reader to Theorem 4.3.1.7 for the precise statement. S consequence of ?? 3 above is the following statement, concerning the properties of 1-categories of formal models for almost perfect complexes on X 2 dAn k : Theorem 4 (Theorem 4.3.3.10). Let X 2 dAn k be a derived k-analytic space and let F 2 Coh + (F) be a bounded below almost perfect complex on X. For any derived formal model X of X, there exists G 2 Coh + (X) and an equivalence G rig ' F. Furthermore, the full subcategory of Coh + (X) ⇥ Coh + (X) Coh + (X) /F spanned by formal models of F is filtered.

?? 4 is another key technical ingredient in the proof of the existence of a plus pushforward construction. The third auxiliary result we need is a refinement of the existence theorem for formal models for morphisms of derived analytic spaces proven in [Ant18a]. It can be stated as follows:

Theorem 5 (Theorem 4.4.0.1). Let f : X ! Y be a flat map between derived k-analytic spaces. Then there are formal models X and Y for X and Y respectively and a flat map f : X ! Y whose generic fiber is equivalent to f .

The classical analogue of ?? 5 was proven by Bosch and Lutkëbohmert in [BLR95b]. The proof of this theorem is not entirely obvious: indeed the algorithm provided in [Ant18a] proceeds by induction on the Postnikov tower of both X and Y , and at each step uses [HPV16b, Theorem 7.3] to choose appropriately formal models for ⇡ i (O X alg) and ⇡ i (O Y alg). In the current situation, however, the flatness requirement on f makes it impossible to freely choose a formal model for ⇡ i (O X alg). We circumvent the problem by proving a certain lifting property for morphisms of almost perfect complexes: Theorem 6 (Theorem 4.3.3.11). Let X 2 dAn k be a derived k-analytic space and let f : F ! G be a morphism in Coh + (X). Let X denote a given formal model for X. Suppose, futhermore, that we are given formal models e F, e G 2 Coh + (X) for F and G, respectively. Then, there exists a non-zero element t 2 m such that the map t n f admits a lift e f : e F ! e G, in the 1-category Coh + (X). Finally, the techniques of the current text allow us to prove the following generalization of [PY18b, Theorem 8.6]:

Theorem 7 (Theorem 4.6.0.3). Let S be a rigid k-analytic space. Let X, Y be rigid k-analytic spaces over S. Assume that X is proper and flat over S and that Y is separated over S. Then the 1-functor Map S (X, Y ) is representable by a derived k-analytic space separated over S.

Notation and conventions

In this paper we freely use the language of 1-categories. Although the discussion is often independent of the chosen model for 1-categories, whenever needed we identify them with quasi-categories and refer to [Lur09c] for the necessary foundational material.

The notations S and Cat 1 are reserved to denote the 1-categories of spaces and of 1-categories, respectively. If C 2 Cat 1 we denote by C ' the maximal 1-groupoid contained in C. We let Cat st 1 denote the 1-category of stable 1-categories with exact functors between them. We also let Pr L denote the 1-category of presentable 1-categories with left adjoints between them. Similarly, we let Pr L st denote the 1-categories of stably presentable 1-categories with left adjoints between them. Finally, we set

Cat st,⌦ 1 := CAlg(Cat st 1 ) , Pr L,⌦ st := CAlg(Pr L st ).
Given an 1-category C we denote by PSh(C) the 1-category of S-valued presheaves. We follow the conventions introduced in [PY16d, §2.4] for 1-categories of sheaves on an 1-site.

For a field k, we reserve the notation CAlg k for the 1-category of simplicial commutative rings over k. We often refer to objects in CAlg k simply as derived commutative rings. We denote its opposite by dA↵ k , and we refer to it as the 1-category of derived affine schemes. We say that a derived ring A 2 CAlg k is almost of finite presentation if ⇡ 0 (A) is of finite presentation over k and ⇡ i (A) is a finitely presented ⇡ 0 (A)-module. 1 We denote by dA↵ afp k the full subcategory of dA↵ k spanned by derived affine schemes Spec(A) such that A is almost of finite presentation. When k is either a non-archimedean field equipped with a non-trivial valuation or is the field of complex numbers, we let An k denote the category of analytic spaces over k. We denote by Sp(k) the analytic space associated to k.

Let k denote a non-archimedean field equipped with a rank 1 valuation. We let k = {x 2 k : |x|  1} denote its ring of integers. We assume that k admits a finitely generated ideal of definition m. (i) Let R be a discrete commutative ring. Let T disc (R) denote the full subcategory of Rschemes spanned by affine spaces A n R . We say that a morphism in T disc (R) is admissible if it is an isomorphism. We endow T disc (R) with the trivial Grothendieck topology.

(ii) Let T ad (k ) denote the full subcategory of k -schemes spanned by formally smooth formal schemes which are topologically finitely generated over k . A morphism in T ad (k ) is said to be admissible if it is formally étale. We equip the category T ad (k ) with the formally étale topology, ⌧ ét.

(iii) Denote T an (k) the category of smooth k-analytic spaces. A morphism in T an (k) is said to be admissible if it is étale. We endow T an (k) with the étale topology, ⌧ ét.

In what follows, we will let T denote either one of the categories introduced above. We let ⌧ denote the corresponding Grothendieck topology. which sends a geometric morphism (f 1 , f ⇤ ) to the functor induced by composition with f 1 . Since the left adjoint of a geometric morphism preserves finite limits, it follows that it respects the full subcategories of Tstructures. In other words, we obtain a well defined functor

Str T : Top R op ! Cat 1 .
This defines a Cartesian fibration p Str : Top

O(U ) O(V ) O 0 (U ) O 0 (V ) O(f ) ↵ U ↵ V O 0 (f )
is a pullback square in X. We denote by Str loc T (X) the (non full) subcategory of Str T (X) spanned by local structures and local morphisms between these. (i) Let R be a discrete commutative ring. A T disc (R)-structure on an 1-topos X is simply a product preserving functor O : T disc (R) ! X. When X = S is the 1-topos of spaces, we can therefore use [Lur09c, Proposition 5.5.9.2] to identify the 1-category Str T disc (R) (X) with the underlying 1category CAlg R of the model category of simplicial commutative R-algebras. It follows that Str T disc (R) (X) is canonically identified with the 1-category of sheaves on X with values in CAlg R . For this reason, we write CAlg R (X) rather than Str loc T disc (R) (X). (ii) Let X denote a formal scheme over k complete along t 2 k . Denote by X fét the small formal étale site on X and denote X := Shv(X ft , ⌧ ét ) ^denote the hypercompletion of the 1-topos of formally étale sheaves on X. We define a T ad (k )-structure on X as the functor which sends U 2 X fét to the sheaf O(U ) 2 X defined by the association

V 2 X fét 7 ! Hom fSch k (V, U ) 2 S. In this case, O(A 1
k ) corresponds to the sheaf of functions on X whose support is contained in the t-locus of X. To simplify the notation, we write fCAlg k (X) rather than Str loc T ad (k ) (X). (iii) Let X be a k-analytic space and denote Xét the associated small étale site on X. Let X := Shv(Xét, ⌧ ét ) denote the hypercompletion of the 1-topos of étale sheaves on X. We can attach to X a T an (k)-structure on X as follows: given U 2 T an (k), we define the sheaf O(U ) 2 X by

Xét 3 V 7 ! Hom An k (V, U ) 2 S.
As in the previous case, we can canonically identify O(A 1 k ) with the usual sheaf of analytic functions on X. We write AnRing k (X) rather than Str loc Tan(k) (X). Construction 4.2.0.5. Let X be an 1-topos. We can relate the 1-categories

Str T disc (k ) (X), Str T disc (k) (X),
Str T ad (k ) (X) and Str Tan(k) (X) as follows. Consider the following functors

(i) the functor ⌦ k k : T disc (k ) ! T disc (k).
induced by base change along the map k ! k.

(ii) The functor These functors respect the classes of admissible morphisms and are continuous morphisms of sites. It follows that precomposition with them induce well defined functors

( ) t : T disc (k ) ! T ad (k ).
Str T disc (k) (X) ! Str T disc (k ) (X) , ( )alg : Str T ad (k ) (X) ! Str T disc (k ) (X) ( ) + : Str Tan(k) (X) ! Str T ad (k ) (X) , ( )alg : Str Tan(k) (X) ! Str T disc (k) (X).
The first functor simply forgets the k-algebra structure to a k -algebra one via the natural map k ! k. We refer to the second and fourth functors as the underlying algebra functors. The third functor is an analogue of taking the subring of power-bounded elements in rigid geometry.

Using the underlying algebra functors introduced in the above construction, we can at last introduce the definitions of derived formal scheme and derived k-analytic space. They are analogous to each other: Definition 4.2.0.6. A T ad (k )-structured 1-topos X := (X, O X ) is said to be a derived formal Deligne-Mumford stackk -stack if there exists a collection of objects {U i } i2I in X such that `i2I U i ! 1 X is an effective epimorphism and the following conditions are met: (i) for every i 2 I, the T ad (k )-structured 1-topos (X /Ui , ⇡ 0 (O X | Ui )) is equivalent to the T ad (k )-structured 1-topos arising from an affine formal k -scheme via the construction given in Theorem 4.2.0.4.

(ii) For each i 2 I and each integer n 0, the sheaf

⇡ n (O X alg| Ui ) is a quasi-coherent sheaf over (X /Ui , ⇡ 0 (O X | Ui )).
We say that X = (X, O X ) is a formal derived k -scheme if it is a derived formal Deligne Mumford stack and furthermore its truncation t 0 (X) := (X, ⇡ 0 (O X )) is equivalent to the T ad (k )-structured 1-topos associated to a formal scheme via Theorem 4.2.0.4.

Definition 4.2.0.7. A T an (k)-structured 1-topos X := (X, O X ) is said to be a derived k-analytic space if X is hypercomplete and there exists a collection of objects {U i } i2I in X such that `i2I U i ! 1 X is an effective epimorphism and the following conditions are met:

(i) for each i 2 I, the T an (k)-structured 1-topos (X /Ui , ⇡ 0 (O X | Ui )
) is equivalent to the T an (k)-structured 1-topos arising from an ordinary k-analytic space via the construction given in Theorem 4.2.0.4.

(ii) For each i 2 I and each integer n 0, the sheaf Following [?, §8.1], we let CAlg ad denote the 1-category of simplicial commutative rings equipped with an adic topology on their 0-th truncation. Morphisms are morphisms of simplicial commutative rings that are furthermore continuous for the adic topologies on their 0-th truncations. We set

⇡ n (O X alg| Ui ) is a coherent sheaf on (X /Ui , O X | Ui ).
CAlg ad k := CAlg ad k / ,
where we regard k equipped with its m-adic topology. Thanks to [Ant18a, Remark 3.1.4], the underlying algebra functor ( )alg : fCAlg k (X) ! CAlg k (X) factors through CAlg ad k (X). We denote by ( ) ad the resulting functor:

( ) ad : fCAlg k (X) ! CAlg ad k (X). Definition 4.2.0.9. Let A 2 fCAlg k (X). We say that A is topologically almost of finite type over k if the underlying sheaf of k -adic algebras A ad is t-complete, ⇡ 0 (Aalg) is sheaf of topologically of finite type k -adic algebras and for each i > 0, ⇡ i (A) is finitely generated as ⇡ 0 (A)-module.

We say that a derived formal Deligne-Mumford stackstack X := (X, O X ) if topologically almost of finite type over k if its underlying 1-topos is coherent (cf. [Lur11f, §3]) and O X 2 fCAlg k (X) is topologically almost of finite type over k . We denote by dfDM taft (resp. dfSch taft ) the full subcategory of dfDM k spanned by those derived formal Deligne-Mumford stackstacks X that are topologically almost of finite type over k (resp. and whose truncation t 0 (X) is equivalent to a formal k -scheme). We refer to this functor as the derived generic fiber functor or as the derived rigidification functor. (ii) The restriction of ( ) rig : dfDM taft ! dAn k to the full subcategory fSch taft k is canonically equivalent to Raynaud's generic fiber functor.

The transformation of pregeometries

(iii) Every derived analytic space X 2 dAn k whose truncation is an ordinary k-analytic space2 lies in the essential image of the functor ( ) rig .

Fix a derived formal Deligne-Mumford stackstack X := (X, O X ) and a derived k-analytic space

Y := (Y, O Y ). We set O X -Mod := O X alg-Mod , O Y -Mod := O Y alg-Mod.
We refer to O X -Mod as the stable 1-category of O X -modules. Similarly, we refer to O Y -Mod as the stable 1-category of O Y -modules. The derived generic fiber functor induces a functor

( ) rig : O X -Mod ! O X rig -Mod.
Definition 4.2.0.11. Let X 2 dfDM k be a derived k -adic Deligne-Mumford stackstack and let X 2 dAn k be a derived k-analytic space. The 1-category Coh + (X) (resp. Coh + (X)) of almost perfect complexes on X (resp. on X) is the full subcategory of O X -Mod (resp. of O X -Mod) spanned by those O X -modules (resp. O X -modules) F such that ⇡ i (F) is a coherent sheaf on t 0 (X) (resp. on t 0 (X)) for every i 2 Z and ⇡ i (F) ' 0 for i ⌧ 0.

For later use, let us record the following result: Proposition 4.2.0.12 ( [?] & [PY18b, Theorem 3.4]). Let X be a derived affine k -adic scheme. Let A := (X; O X alg). Then the functor (X; ) restricts to

Coh + (X) ! Coh + (A)
and furthermore this is an equivalence. Similarly, if X is a derived k-affinoid space, 3 and B := (X; O X alg), then (X; ) restricts to

Coh + (X) ! Coh + (B)
, and furthermore this is an equivalence.

To complete this short review, we briefly discuss the notion of the k -adic and k-analytic cotangent complexes. The two theories are parallel, and for sake of brevity we limit ourselves to the first one. We refer to the introduction of [PY17b] for a more thorough review of the k-analytic theory.

In [Ant18a, §3.4] it was constructed a functor ⌦ 1 ad : O X -Mod ! fCAlg k (X) /O X , which we refer to as the k -adic split square-zero extension functor. Given F 2 O X -Mod, we often write O X F instead of ⌦ 1 ad (F). Remark 4.2.0.13. Although the 1-category O X -Mod is not sensitive to the T ad (k )-structure on O X , the functor

⌦ 1
ad depends on it in an essential way. Definition 4.2.0.14. The functor of k -adic derivations is the functor

Der ad k (X; ) : O X -Mod ! S defined by Der ad k (X; F) := Map fCAlg k (X) /O X (O X , O X F).
For formal reasons, the functor Der ad k (X; ) is corepresentable by an object L ad X 2 O X -Mod. We refer to it as the k -adic cotangent complex of X. The following theorem summarizes its main properties: Theorem 4.2.0.15 ( [Ant18a, Proposition 3.4.4, Corollary 4.3.5, Proposition 3.5.8]). Let X := (X, O X ) be a derived k -adic Deligne-Mumford stackstack. Let t n X := (X, ⌧ n O X ) be the n-th truncation of X. Then:

(i) the k -adic cotangent complex L ad X belongs to Coh + (X);

(ii) in Coh + (X rig
) there is a canonical equivalence

(L ad X ) rig ' Lan X rig ,
where Lan X rig denotes the analytic cotangent complex of the derived k-analytic space X rig ;

(iii) the algebraic derivation classifying canonical map (X,

⌧ n+1 O X ) ! (X, ⌧ n O X ) can be canonically lifted to a k -adic derivation L ad t n X ! ⇡ n+1 (O X )[n + 2].

Formal models for almost perfect complexes 4.3.1 Formal descent statements

We assume that k admits a finitely generated ideal of definition m. We also fix a set of generators t 1 , . . . , t n 2 m. We start by recalling the notion of m-nilpotent almost perfect complexes.

Definition 4.3.1.1. Let X be a derived k -adic Deligne-Mumford stackstack topologically almost of finite presentation. We let Coh + nil (X) denote the fiber of the generic fiber functor (4.1.0.1):

Coh + nil (X) := fib ✓ Coh + (X) ( ) rig ! Coh + (X rig ) ◆ .
We refer to Coh + nil (X) as the full subcategory of m-nilpotent almost perfect complexes on X.

A morphism f : X ! Y in dfDM taft k induces a commutative diagram Coh + (Y) Coh + (X) Coh + (Y rig ) Coh + (X rig ). f ⇤ ( ) rig ( ) rig (f rig ) ⇤ (4.3.1.1)
In particular, we see that f ⇤ preserves the subcategory of m-nilpotent almost perfect complexes on X. Moreover, as both Coh + (X) and Coh + (X rig ) satisfy étale descent, we conclude that Coh + nil (X) satisfies étale descent as well. (X) is m-nilpotent if and only if for every i 2 Z the coherent sheaf ⇡ i (F) is annihilated by some power of the ideal m.

Proof. The question is étale local on X. In particular, we can assume that X is a derived formal affine scheme topologically of finite presentation. Write

A := (X, O X alg). Let X := X rig . Then [Ant18a, Corollary 4.1.3] shows that t 0 (X rig
) ' (t 0 (X)) rig .

In particular, we deduce that X is a derived k-affinoid space. Write B := (X, O X alg).

(ii) The functor ( ) rig : Coh + (X) ! Coh + (X rig ) factors as

⇤ : Coh + loc (X) ! Coh + (X rig ).
Moreover, the essential images of ( ) rig and ⇤ coincide.

(iii) If X is affine, then the functor ⇤ is an equivalence.

Proof. We start by proving [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. Using [HPV16b, Corollary 2.9] we have to check that the t-structure on Coh + (X) restricts to a t-structure on Coh + nil (X) and that the inclusion

i : Coh ñil (X) , ! Coh ~(X)
admits a right adjoint R whose counit i(R(X)) ! X is a monomorphism for every X 2 Coh ~(X). For the first statement, we remark that it is enough to check that the functor (

) rig : Coh + (X) ! Coh + (X rig ) is t-exact. As both Coh + (X) and Coh + (X rig
) satisfy étale descent in X, we can test this locally on X. When X is affine, the assertion follows directly from Theorem 4.2.0.12. As for the second statement, we first observe that

Coh ~(X) ' Coh ~(t 0 (X)).
We can therefore assume that X is underived. At this point, the functor R can be explicitly described as the functor sending F 2 Coh ~(X) to the subsheaf of F spanned by m-nilpotent sections. The proof of ( 1) is thus complete.

We now turn to the proof of (2). The existence of ⇤ and the factorization ( ) rig ' ⇤ L follow from the definitions. Moreover, L : Coh + (X) ! Coh + loc (X) is essentially surjective (cf. [HPV16b, Lemma 2.3]). It follows that the essential images of ( ) rig and of ⇤ coincide.

Finally, (3) follows directly from Theorem 4.2.0.12 and [HPV16b, Theorem 2.12].

The commutativity of (4.3.1.1) implies that a morphism f : X ! Y in dfDM taft k induces a well defined functor

f ⇤ : Coh + loc (Y) ! Coh + loc (X).
It is a simple exercise in 1-categories to promote this construction to an actual functor

Coh + loc : dfDM taft k op ! Cat st 1 .
Having Theorem 4.

f ⇤ • : Coh + loc (X) ! lim [n]2 Coh + loc (U • ) (4.3.1.2)
is an equivalence. Using [HPV16b, Lemma 3.20] we can endow the right hand side with a canonical t-structure.

It follows from the characterization of the t-structure on Coh + loc (X) given in Theorem 4.3.1.6 that f ⇤ • is t-exact. We will prove in Theorem 4.3.2.4 that f ⇤

• is fully faithful. Assuming this fact, we can complete the proof as follows. We only need to check that f ⇤

• is essentially surjective. Let C be the essential image of f ⇤ • . We now make the following observations:

(i) the heart of lim Coh + loc (U • ) is contained in C. Indeed, Theorem 4.3.1.4 implies that ⇤ n : Coh + loc (U n ) ! Coh + (U rig n )
is an equivalence. These equivalences induce a t-exact equivalence

Coh + (X rig ) ' lim [n]2 Coh + loc (U • ). (4.3.1.3)
Passing to the heart and using the canonical equivalences

Coh loc (X) ' Coh loc (t 0 (X)) , Coh ~(X rig ) ' Coh ~(t 0 (X rig )),
we can invoke the classical Rayanaud's theorem on formal models of coherent sheaves to deduce that the heart of the target of f ⇤ • is contained in its essential image.

(ii) The subcategory C is stable. Indeed, let

F 0 F F 00 '
be a fiber sequence in Coh + (X rig ) ' lim Coh + loc (U • ) and suppose that two among F, F 0 and F 00 belong to C. Without loss of generality, we can assume that F and F 00 belong to C. Then choose elements F and F 00 in Coh + loc (X) representing F and F 00 . Since f ⇤ • is fully faithful, we can find a morphism e : F ! F 00 lifting . Set F 0 := fib( e : F ! F 00 ).

Then ⇤(F 0 ) ' F 0 , which means that under the equivalence (4.3.1.3) the object F 0 belongs to C.

These two points together imply that f ⇤ • is essentially surjective on cohomologically bounded elements. As both the t-structures on source and target of f ⇤

• are left t-complete, the conclusion follows. and assume moreover that X is quasi-compact and quasi-separated. Then the canonical map

⇤ : Coh + loc (X) ! Coh + (X rig )
introduced in Theorem 4.3.1.6 is an equivalence.

Proof. Let f • : U • ! X be a derived affine k -adic Zariski hypercover. Consider the induced commutative diagram

Coh + loc (X) lim [n]2 Coh + loc (U n ) Coh + (X rig ) lim [n]2 Coh + (U rig n ), f ⇤ • ⇤ ⇤• f ⇤ • where we set f • := (f • ) rig .
The right vertical map is an equivalence thanks to Theorem 4.3.1.6. On the other hand, Coh + (X rig ) satisfies descent in X, and therefore the bottom horizontal map is also an equivalence. Finally, Theorem 4.3.1.7 implies that the top horizontal map is an equivalence as well. We thus conclude that

⇤ : Coh + loc (X) ! Coh + (X rig
) is an equivalence.

Corollary 4.3.1.9. Let X 2 dfSch taft k and assume moreover that it is quasi-compact and quasi-separated. For any F 2 Coh + (X rig ), the 1-category FM(F) is non-empty.

Proof. The localization functor L : Coh + (X) ! Coh + loc (X) is essentially surjective by construction. Since X is a quasi-compact and quasi-separted derived k -adic scheme topologically of finite presentation, Theorem 4.3.1.8 implies that ⇤ :

Coh + loc (X) ! Coh + (X rig
) is an equivalence. The conclusion follows.

Proof of Theorem 4.3.1.7: fully faithfulness

The only missing step in the proof of Theorem 4.3.1.7 is the full faithfulness of the functor (4.3.1.2). We will address this question by passing to the 1-categories of ind-objects. Let X be a quasi-compact and quasi-separated derived k -adic scheme locally topologically almost of finite presentation.

f : U ! X be a formally étale morphism. Then f induces a commutative diagram

Ind(Coh + (X)) Ind(Coh + loc (X)) Ind(Coh + (U)) Ind(Coh + loc (U)). f ⇤ L X f ⇤ L U
The functors f ⇤ and f ⇤ commute with colimits, and therefore they admit right adjoints f ⇤ and f ⇤ . In particular, we obtain a Beck-Chevalley transformation

✓ : L X f ⇤ ! f ⇤ L U . (4.3.2.1)
A key step in the proof of the full faithfulness of the functor (4.3.1.2) is to verify that ✓ is an equivalence when evaluated on objects in Coh ~(U). Let us start with the following variation on [HPV16b, Lemma 7.14]:

Lemma 4.3.2.1. Let K C C Q C K D D Q D i C F K L C F F Q i D L D (4.3.2.2)
be a diagram of stable 1-categories and exact functors between them. Assume that:

(i) the functors i C and i D are fully faithful and admit right adjoints R C and R D , respectively;

(ii) the functors L C and L D admit fully faithful right adjoints j C and j D , respectively;

(iii) the rows are fiber and cofiber sequences in Cat st 1 ;

(iv) the functors F , F K and F Q admit right adjoints G, G K and G Q , respectively.

Let X 2 D be an object. Then the following statements are equivalent:

(i) the Beck-Chevalley transformation

q X : L C (G(X)) ! G Q (L D (X))
is an equivalence;

(ii) the Beck-Chevalley transformation

 R D (X) : i C (G K (R D (X))) ! G(i D (R D (X)))
is an equivalence.

Proof. Since j C and i C are fully faithful, it is equivalent to check that

j C (L C (G(X))) ! j C (G Q (L D (X)))
is an equivalence if and only if  R D (X) is an equivalence. Using the natural equivalences

j C G ' Gj D , G K R D ' R C G we obtain the following commutative diagram i C (R C (G(X))) G(X) j C (L C (G(X))) G(i D (R D (X))) G(X) G(j D (L D (X))).
Moreover Proof. Using Theorem 4.3.2.1, we see that it is enough to prove that the Beck-Chevalley transformation associated to the square

Ind(Coh + nil (X)) Ind(Coh + (X)) Ind(Coh + nil (U)) Ind(Coh + (U)) f ⇤ f ⇤
is an equivalence when evaluated on objects of Coh ñil (U). As the horizontal functors are fully faithful, it is enough to check that the functor

f ⇤ : Ind(Coh + (U )) ! Ind(Coh + (X)) takes Coh ñil (U) to Ind(Coh + nil (X)). Let F 2 Coh ñil (U).
We have to verify that (f ⇤ (F)) rig ' 0. Since F is coherent and in the heart and since U is quasi-compact we see that there exists an element a 2 m such that the map

µ a : F ! F given by multiplication by a is zero. Therefore f ⇤ (µ a ) : f ⇤ (F) ! f ⇤ (F) is homotopic to zero. Since f ⇤ (µ a ) is equivalent to the endomorphism f ⇤ (F)
given by multiplication by a, we conclude that (f ⇤ (F)) rig ' 0. The conclusion follows.

Having these adjointability statements at our disposal, we turn to the actual study of the full faithfulness of the functor (4.3.1.2). Let U • : op ! dfSch taft k be an affine k -adic Zariski hypercovering of X and let f • : U • ! X be the augmentation morphism. The morphism

f • induces functors f ⇤ • : Ind(Coh + (X)) ! lim [n]2 Ind(Coh + (U n )) and f ⇤ • : Ind(Coh + loc (X)) ! lim [n]2 Ind(Coh + loc (U n ))
. These functors commute by construction with filtered colimits, and therefore they admit right adjoints, that we denote respectively as

f •⇤ : lim [n]2 Ind(Coh + (U n )) ! Ind(Coh + (X)) and f •⇤ : lim [n]2 Ind(Coh + loc (U n )) ! Ind(Coh + loc (X)).

Moreover, the functors f ⇤

• and f ⇤ • fit in the following commutative diagram:

Ind(Coh + (X)) lim [n]2 Ind(Coh + (U • )) Ind(Coh + loc (X)) lim [n]2 Ind(Coh + loc (U • )). L f ⇤ • L• f ⇤ •
In particular, we have an associated Beck-Chevalley transformation

✓ : L f •⇤ ! f •⇤ L • . ( 4
f •⇤ : lim [n]2 Ind(Coh + (U n )) ! Ind(Coh + (X))
with the functor informally described by sending a descent datum

F • 2 lim Ind(Coh + (U • )) to lim [n]2 f n⇤ F n 2 Ind(Coh + (X)).
Similarly, the functor f •⇤ sends a descent datum

F • 2 lim Ind(Coh + loc (U • ) to lim [n]2 f n⇤ F n 2 Ind(Coh + loc (X)
). We therefore have to show that the Beck-Chevalley transformation

✓ : L ✓ lim [n]2 f n⇤ F n ◆ ! lim [n]2 f n⇤ (L n F n )
is an equivalence whenever each F n belongs to Coh ~(U n ). First notice that the functors f •⇤ and f •⇤ are left t-exact. In particular, if

F • 2 lim Ind(Coh ~(U • )) then both Lf •⇤ (F • ) and f •⇤ (F • ) are coconnective.
As the t-structures on lim Ind(Coh + (U • )) and on lim Ind(Coh + loc (U • )) are right t-complete, we conclude that it is enough to prove that ⇡ i (✓) is an isomorphism for every i 2 Z. We now observe that for m i + 2 we have

⇡ i ✓ lim [n]2 f n⇤ (L n F n ) ◆ ' ⇡ i ✓ lim [n]2 m f n⇤ (L n F n ) ◆ ,
and similarly

⇡ i ✓ L ✓ lim [n]2 f n⇤ F n ◆◆ ' L ✓ ⇡ i ✓ lim [n]2 f n⇤ F n ◆◆ ' L ✓ ⇡ i ✓ lim [n]2 m f n⇤ F n⇤ ◆◆ .
It is therefore enough to prove that for every m 0 the canonical map

L ✓ lim [n]2 m f n⇤ F n ◆ ! lim [n]2 m f n⇤ (L n F n )
is an equivalence. As L commutes with finite limits, we are reduced to show that the canonical map

L(f n⇤ F n ) ! f n⇤ (L n F n ) is an equivalence whenever F n 2 Coh ~(U n ), which follows from Theorem 4.3.2.2.
Corollary 4.3.2.4. Let X and f • : U • ! X be as in the above discussion. Then the functor

f ⇤ • : Coh + loc (X) ! lim [n]2 Coh + loc (U n ) is fully faithful.

Proof. As the functor f ⇤

• is t-exact and the t-structure on both categories is left complete, we see that it is enough to reduce ourselves to prove that f ⇤

• is fully faithful when restricted to Coh b loc (X). Consider the following commutative cube:

Coh + (X) lim [n]2 Coh + (U n ) Ind Coh + (X) lim [n]2 Ind Coh + (U n ) Coh + loc (X) lim [n]2 Coh + loc (U n ) Ind Coh + loc (X) lim [n]2 Ind Coh + loc (U n ) . f ⇤ • f ⇤ • L X L U• f ⇤ • f ⇤ • (4.3.2.4)
First of all, we observe that the diagonal functors are all fully faithful. It is therefore enough to prove that the functor f ⇤ • : Ind(Coh + loc ((X)) ! lim [n]2 Ind(Coh + loc (U n )) is fully faithful when restricted to Coh + loc (X). As this functor admits a right adjoint f •⇤ , it is in turn enough to verify that for every F 2 Coh b loc (F) the unit transformation

⌘ : F ! f •⇤ f ⇤ • (F)
is an equivalence. Proceeding by induction on the number of nonvanishing homotopy groups of F, we see that it is enough to deal with the case of F 2 Coh loc (F).

As the functor L X : Coh + (X) ! Coh + loc (X) is essentially surjective and t-exact, we can choose F 2 Coh ~(X) and an equivalence L X (F) ' F. Moreover, the unit transformation

F ! f •⇤ f ⇤ • F is an equivalence.
It is therefore enough to check that the Beck-Chevalley transformation associated to the front square is an equivalence when evaluated on objects in lim Coh ~(U n ). This is exactly the content of Theorem 4.3.2.3.

Categories of formal models

Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. We established in Theorem 4.3.1.9 that for any F 2 Coh + (X rig ) the 1-category of formal models FM(F) is non-empty. Actually, we can use Theorem 4.3.1.8 to be more precise about the structure of FM(F). We are in particular interested in showing that it is filtered. We start by recording the following immediate consequence of Theorem 4.3.1.8: admits a right adjoint j : Ind(Coh + (X rig )) ! Ind(Coh + (X)), which is furthermore fully faithful.

Proof. Theorem 4.3.1.8 implies that the functor ( ) rig induces the equivalence

⇤ : Coh + loc (X) ⇠ ! Coh + (X rig ).
In other words, we see that the diagram

Coh + nil (X) Coh + (X) 0 Coh + (X rig ) ( ) rig is a pushout diagram in
Hom X (F, G) ⌦ k k ' 0.
In other words, Hom X (F, G) is m-nilpotent in Mod k .

Proof. Since X is quasi-compact, we can find a finite formal Zariski cover U i = Spf(A i ) by formal affine schemes. Let U • be the Čech nerve. Since this is a formal Zariski cover, there exists m 0 such that

Hom X (F, G) ' lim [n]2 m Hom Un (F| Un , G| Un ).
Since the functor ⌦ k k : Mod k ! Mod k is exact, it commutes with finite limits. Therefore, we see that it is enough to prove that the conclusion holds after replacing X by U m . Since X is quasi-compact and quasi-separated, we see that each U m is quasi-compact and separated. In other words, we can assume from the very beginning that X is quasi-compact and separated. In this case, each U m will be formal affine, and therefore we can further reduce to the case where X is formal affine itself.

Assume therefore X = Spf(A). In this case, Coh + (X) ' Coh + (A) lives fully faithfully inside Mod A . Notice that A ! A ⌦ k k is a Zariski open immersion. Therefore,

Hom A (F, G) ⌦ k k ' Hom A (F, G) ⌦ A (A ⌦ k k) ' Hom A (F ⌦ A k , G ⌦ A k ) ' 0.
Thus, the proof is complete.

Corollary 4.3.3.4. Let X be as in the previous lemma. Given F, G 2 Coh + (X), the canonical map

Hom X (F, G) ⌦ k k ! Hom X rig (F rig , G rig )
is an equivalence.

Proof. Denote by R : Ind(Coh + (X)) ! Ind(Coh + nil (X)) the right adjoint to the inclusion

i : Ind(Coh + nil (X)) ,! Ind(Coh + (X)).
Then for any G 2 Coh + (X) we have a fiber sequence

iR(G) ! G ! j(G rig ).
In particular, we obtain a fiber sequence

Hom X (F, iR(G)) ! Hom X (F, G) ! Hom X (F, j(G rig )).

Now observe that

Hom X (F, j(G rig )) ' Hom X rig (F rig , G rig ). Notice also that since k ! k is an open Zariski immersion, Hom X rig (F rig , G rig ) ⌦ k k ' Hom X rig (F rig , G rig
). In particular, applying ⌦ k k : Mod k ! Mod k we find a fiber sequence

Hom X (F, iR(G)) ⌦ k k ! Hom X (F, G) ⌦ k k ! Hom X rig (F rig , G rig ).
It is therefore enough to check that Hom X (F, iR(G)) ⌦ k k ' 0. Since i is a left adjoint, we can write

iR(G) ' colim ↵2I G ↵ ,
where I is filtered and G ↵ 2 Coh + nil (X). As F is compact in Ind(Coh + (X)), we find

Hom X (F, iR(G)) ⌦ k k ' ✓ colim ↵2I Hom X (F, G ↵ ) ◆ ⌦ k k ' colim ↵2I Hom X (F, G ↵ ) ⌦ k k.
Since each G ↵ belongs to Coh + nil (X), Theorem 4.3.3.3 implies that Hom X (F, G ↵ ) ⌦ k k ' 0. The conclusion follows. Construction 4.3.3.6. Choose generators t 1 , . . . , t n for m. We consider N n as a poset with order given by

(m 1 , . . . , m n )  (m 0 1 , . . . , m 0 n ) () m 1  m 0 1 , m 2  m 0 2 , . . . , m n  m 0 n
Introduce the functor

K : N n ! Ind(Coh ~(Spf (k )))
defined as follows: K sends every object to k , and it sends the morphism m  m 0 to multiplication by t m 0 m . By abuse of notation, we still denote the composition of K with the inclusion Ind(Coh ~(k )) ! Ind(Coh + (k )) by K.

Let now X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + (X). The natural morphism q : X ! Spf(k ) induces a functor

q ⇤ : Ind(Coh + (Spf(k ))) ! Ind(Coh + (X)).
We define the functor K F as

K F := q ⇤ (K( )) ⌦ F : N n ! Ind(Coh + (X)).
We let F loc denote the colimit of the functor

K F . Let G 2 Coh + (X rig
) and let ↵ : F rig ! G be a given map. Notice that the natural map

F rig ! colim N n (K F ( )) rig
is an equivalence. Therefore ↵ induces a cone

(K F ( )) rig ! G,
which is equivalent to the given of a cone

K F ( ) ! j(G).
Specializing this construction for ↵ = id F rig , we obtain a canonical map

F : F loc ! j(F rig ).
Lemma 4.3.3.7. Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + nil (X). Then F loc ' 0. Proof. For any G 2 Coh + (X), we write Hom X (G, F) 2 Mod k for the k -enriched mapping space. As G is compact in Ind(Coh + (X)), we have

Hom X (G, F loc ) ' colim N n Hom X (G, K F ( )) ' Hom X (G, F) ⌦ k k. Theorem 4.3.3.4 implies that Hom X (G, F) ⌦ k k ' Hom X rig (G rig , F rig ) ' 0.
It follows that F loc ' 0. Proof. It is enough to prove that for every i 0 we have

⇡ i Map Ind(Coh + (X)) (G, F loc ) ' 0.
Up to replacing F by F[i], we see that it is enough to deal with the case i = 0. Let therefore ↵ : G ! F loc be a representative for an element in ⇡ 0 Map

Ind(Coh + (X)) (G, F loc ).
As G is compact in Ind(Coh + (X)), the map ↵ factors as ↵ 0 : G ! F, and therefore it induces a map e ↵ : G loc ! F loc making the diagram

G F G loc F loc ↵ 0 e ↵
commutative, where both compositions are equivalent to ↵. Now, Theorem 4.3.3.7 implies that G loc ' 0, and therefore ↵ is nullhomotopic, completing the proof.

Lemma 4.3.3.9. Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + (X). Then the canonical map

F : F loc ! j(F rig ) is an equivalence. Proof. Let G 2 Coh + nil (X). Then Map Ind(Coh + (X)) (G, j(F rig )) ' Map Ind(Coh + (X rig )) (G rig , F) ' 0.
Theorem 4.3.3.8 implies that the same holds true replacing j(F rig ) with F loc . As Coh + nil (X) is a stable full subcategory of Coh + (X), it follows that Hom X (G, j(F)) ' Hom X (G, F loc ) ' 0.

Let H := fib( F ). Then for any G 2 Coh + nil (X), one has Hom X (G, H) ' 0.

On the other hand, H rig ' fib( rig F ) ' 0. It follows that H 2 Ind(Coh + nil (X)), and hence that H ' 0. Thus, F is an equivalence.

Theorem 4.3.3.10. Let X 2 dfSch k be a quasi-compact and quasi-separated derived k -adic scheme. Let F 2 Coh + (X rig ). Then the 1-category FM(F) of formal models for F is non-empty and filtered.

Proof. We know that FM(F) is non-empty thanks to Theorem 4.3.1.9. Pick one formal model F 2 FM(F). Then Theorem 4.3.3.9 implies that the canonical map

F : F loc ! j(F rig ) ' j(F)
is an equivalence. We now observe that FM(F) is by definition a full subcategory of

Coh + (X) /F := Coh + (X) ⇥ Ind(Coh + (X)) Ind(Coh + (X)) /j(F) .
As this 1-category is filtered, it is enough to prove that every object G 2 Coh + (X) /F admits a map to an object in FM(F). Let ↵ : G ! j(F) be the structural map. Using the equivalence F and the fact that G is compact in Ind(Coh + (X)), we see that ↵ factors as G ! F, which belongs to FM(F) by construction.

Corollary 4.3.3.11. Let X 2 dAn k and f : F ! G be a morphism Coh + (X). Suppose we are given a formal model X for X together with formal models F, G 2 Coh + (X) for F and G, respectively. Then there exists a morphism f :

F 0 ! G 0 in the 1-category Coh + (X) lifting t m1 1 . . . t mn n f : F ! G, in Coh + (X)
for suitable non-negative integers m 1 , . . . , m n 0.

Proof. Any map F ! G induces a map F ! j(F) ! j(G). Using the equivalence j(G) ' G loc and the fact that F is compact in Ind(Coh + (X)), we see that the map F ! j(G) factors as F ! G. Unraveling the definition of the functor K G ( ), we see that the conclusion follows.

For later use, let us record the following consequence of Theorem 4.3.3.9: Corollary 4.3.3.12. Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + (X). Then F is m-nilpotent if and only if F loc ' 0. Proof. If F is m-nilpotent, the conclusion follows from Theorem 4.3.3.7. Suppose vice-versa that F loc ' 0. Then Theorem 4.3.3.9 implies that j(F rig ) ' F loc ' 0. Now, Theorem 4.3.3.1 shows that j is fully faithful. In particular it is conservative and therefore F rig ' 0. In other words, F belongs to Coh + nil (X).

Flat models for morphisms of derived analytic spaces

Using the study of formal models for almost perfect complexes carried out in the previous section, we can prove the following derived version of [BL93b, Theorem 5.2]:

Theorem 4.4.0.1. Let f : X ! Y be a proper map of quasi-paracompact derived k-analytic spaces. Assume that:

(i) the truncations of X and Y are k-analytic spaces. 4 (ii) The map f is flat.

Then there exists a proper flat formal model f : X ! Y in dfSch taft k for f .

Proof. We construct, by induction on n, the following data:

(i) derived k -adic schemes X n and Y n equipped with equivalences

X rig n ' t n (X), Y rig n ' t n (Y ).
(ii) Morphisms X n ! X n 1 and Y n ! Y n 1 exhibiting X n 1 and Y n 1 as (n 1)-truncations of X n and Y n , respectively.

(iii) A proper flat morphism f n : X n ! Y n and homotopies making the cube

X rig n Y rig n X rig n 1 Y rig n 1 t n (X) t n (Y ) t n 1 (X) t n 1 (Y ) f rig n f rig n 1
commutative. 4 As opposed to k-analytic Deligne-Mumford stackstacks.

108

Having these data at our disposal, we set

X := colim n X n , Y := colim n Y n ,
and we let f : X ! Y be map induced by the morphisms f n . The properties listed above imply that f is proper and flat and that its generic fiber is equivalent to f . We are therefore left to construct the data listed above. When n = 0, we can apply the flattening technique of Raynaud-Gruson (see [BL93b,Theorem 5.2]) to produce a proper flat formal model f 0 : X 0 ! Y 0 for t 0 (f ) : t 0 (X) ! t 0 (Y ). Assume now that we constructed the above data up to n and let us construct it for n + 1.

Set F := ⇡ n+1 (O X )[n + 2] and G := ⇡ n+1 (O Y )[n + 2].
f ⇤ n L ad Yn f ⇤ n G L ad Xn F t m f ⇤ n ↵ commutative. Set e := t m : L ad Yn ! G.
Then e ↵ and e induce a commutative square

X n [F] X n Y n [G] Y n . d ↵ fn d ˜ (4.4.0.2)
We now define X n+1 and Y n+1 as the square-zero extensions associated to e ↵ and e . In other words, they are defined by the following pushout diagrams:

X n [F] X n X n X n+1 d ↵ d0 , Y n [G] Y n Y n Y n+1 . d0 d ˜
The commutativity of (4.4.0.2) provides a canonical map f n+1 : X n+1 ! Y n+1 , which is readily verified to be proper and flat. We are therefore left to verify that f n+1 is a formal model for f n+1 . Unraveling the definitions, we see that it is enough to produce equivalences a :

(t n X)[F] ⇠ ! (t n X)[F] and b : (t n Y )[G] ⇠ ! (t n Y )[G] making the following diagrams (t n X)[F] t n X (t n X)[F] t n X d t m ↵ a d↵ , (t n Y )[G] t n Y (t n Y )[G] t n Y d t m b d (4.4.0.3)
commutative. The situation is symmetric, so it is enough to deal with t n X. Consider the morphism

t m : F ! F,
which exists because all the elements t i 2 m are invertible in k. For the same reason it is an equivalence, with inverse given by multiplication by t m . This morphism induces a map

a : (t n X)[F] ! (t n X)[F],
which by functoriality is an equivalence. We now observe that the commutativity of (4.4.0.3) is equivalent to the commutativity of Lan t n X F Lan t n X F,

t m ↵ t m
↵ which is immediate. The proof is therefore achieved.

The plus pushforward for almost perfect sheaves

Let f : X ! Y be a proper map between derived k-analytic spaces of finite tor-amplitude. In [PY18b, Definition 7.9] it is introduced a functor

f + : Perf(X) ! Perf(Y ),
and it is shown in Proposition 7.11 in loc. cit. that for every G 2 Coh + (Y ) there is a natural equivalence

Map Coh + (X) (F, f ⇤ G) ' Map Coh + (Y ) (f + (F), G).
In this section we extend the definition of f + to the entire Coh + (X), at least under the stronger assumption of f being flat. Remark 4.5.0.1. In algebraic geometry, the extension of f + to Coh + (X) passes through the extension to QCoh(X) ' Ind(Perf(X)). This is ultimately requires being able to describe every element in Coh + (X) as a filtered colimit of elements in Perf(X), which in analytic geometry is possible only locally.

Therefore, this technique cannot be applied in analytic geometry. When dealing with non-archimedean analytic geometry, formal models can be used to circumvent this problem. Proposition 4.5.0.2. Let f : X ! Y be a proper map between derived k -adic schemes. Assume that f has finite tor amplitude. Then the functor

f ⇤ : Coh + (Y) ! Coh + (X) admits a left adjoint f + : Coh + (X) ! Coh + (Y).
Proof. Let X n := X⇥ Spf(k ) Spec(k /m n ) and define similarly Y n . Let f n : X n ! Y n be the induced morphism. Then by definition of k -adic schemes,

X ' colim n2N X n , Y ' colim n2N Y n ,
and therefore

Coh + (X) ' lim n2N Coh + (X n ), Coh + (Y) ' lim n2N Coh + (Y n ).
Combining [?, Remark 6.4.5.2(b) & Proposition 6.4.5.4(1)], we see that each functor

f ⇤ n : Coh + (Y n ) ! Coh + (X n )
admits a left adjoint f n+ . Moreover, Proposition 6.4.5.4(2) in loc. cit. implies that these functors f n+ can be assembled into a natural transformation, and that therefore they induce a well defined functor

f + : Coh + (X) ! Coh + (Y). Now let F 2 Coh + (X) and G 2 Coh + (Y).
Let F n and G n be the pullbacks of F and G to X n and Y n , respectively. Then

Map Coh + (X) (F, f ⇤ (G)) ' lim n2N Map Coh + (Xn) (F n , f ⇤ n (G n )) ' lim n2N Map Coh + (Yn) (f n+ (F n ), G n ) ' Map Coh + (Y) (f + (F), G),
which completes the proof. Corollary 4.5.0.3. Let f : X ! Y be a proper map between derived analytic spaces. Assume that f is flat. Then the functor

f ⇤ : Coh + (Y ) ! Coh + (X) admits a left adjoint f + : Coh + (X) ! Coh + (Y ).
Proof. Using Theorem 4.4.0.1, we can choose a proper flat formal model f : X ! Y for f . Thanks to Theorem 4.5.0.2, we have a well defined functor

f + : Coh + (X) ! Coh + (Y).
We claim that it restricts to a functor

f + : Coh + nil (X) ! Coh + nil (Y). Using Theorem 4.3.3.12, it is enough to prove that f + (F) loc ' 0. Extending f + to a functor f + : Ind(Coh + (X)) ! Ind(Coh + (Y)), we see that f + (F) loc ' f + (F loc ) ' 0.
Using Theorem 4.3.1.8, we get a well defined functor

f + : Coh + (X) ! Coh + (Y ).
We only have to prove that it is left adjoint to f ⇤ . Let F 2 Coh + (X) and G 2 Coh + (Y ). Choose a formal model F 2 Coh + (X). Then unraveling the construction of f + , we find a canonical equivalence

f + (F) ' f + (F) rig .
We now have the following sequence of natural equivalences:

Map Coh + (Y ) (f + (F), G) ' Map Coh + (Y ) ((f + (F)) rig , G rig ) ' Map Coh + (X) (f + (F), G) ⌦ k k by Theorem 4.3.3.4 ' Map Coh + (X) (F, f ⇤ G) ⌦ k k ' Map Coh + (X) (F rig , (f ⇤ G) rig ) by Theorem 4.3.3.4 ' Map Coh + (X) (F, f ⇤ G).
The proof is therefore complete.

g + (q ⇤ (F)) ! p ⇤ (f + (F))
is an equivalence. Choose also a formal model F 2 Coh + (X) for F. It is then enough to prove that the canonical map

g + (q ⇤ (F)) ! p ⇤ (f + (F))
is an equivalence. This follows at once by [?, Proposition 6.4.5.4(2)].

Representability of RHilb(X)

Let p : X ! S be a proper and flat morphism of underived k-analytic spaces. We define the functor RHilb(X/S) : dAfd S op ! S by sending T ! S to the space of diagrams

Y T ⇥ S X T i q T p T (4.6.0.1)
where i is a closed immersion of derived k-analytic spaces, and q T is flat.

Proposition 4.6.0.1. Keeping the above notation and assumptions, RHilb(X/S) admits a global analytic cotangent complex.

Proof. Let x : T ! RHilb(X/S) be a morphism from a derived k-affinoid space T 2 dAfd S . It classifies a diagram of the form (4.6.0.1). Unraveling the definitions, we see that the functor Der an RHilb(X/S),x (T ; ) : Coh + (T ) ! RHilb(X/S) can be explicitly written as

Der an RHilb(X/S),x (T ; F) ' Map Coh + (Y ) (Lan Y /T ⇥ S X , q ⇤ T (F)).
Since q T : Y ! T is proper and flat, Theorem 4.5.0.3 implies the existence of a left adjoint q T + : Coh + (Y ) ! Coh + (T ) for q ⇤ T . Moreover, [PY17b, Corollary 5.40] implies that Lan Y /T ⇥ S X 2 Coh + 0(Y ). Therefore, we find

Der an

RHilb(X/S),x (T ; F) ' Map

Coh + (T ) (q T + (Lan Y /T ⇥ S X ), F), and therefore RHilb(X/S) admits an analytic cotangent complex at x. Using Theorem 4.5.0.4, we see that it admits as well a global analytic cotangent complex. Theorem 4.6.0.2. Let X be a k-analytic space. Then RHilb(X) is a derived analytic space. As a second concluding applications, let us mention that the theory of the plus pushforward developed in this paper allows to remove the lci assumption in [PY18b, Theorem 8.6]: Theorem 4.6.0.3. Let S be a rigid k-analytic space. Let X, Y be rigid k-analytic spaces over S. Assume that X is proper and flat over S and that Y is separated over S. Then the 1-functor Map S (X, Y ) is representable by a derived k-analytic space separated over S.

Coherent dualizing sheaves

It should be possible to apply the formalism of this paper to get a reasonable construction for the dualizing sheaf of a morphism of derived k-analytic schemes. Definition 4.7.0.1. Let f : X ! Y be a morphism of derived k-analytic schemes. Choose a formal model f : X ! Y and let ! X/Y be a dualizing sheaf. We set

! X/Y := (! X/Y ) rig .
Proposition 4.7.0.2. Suppose f : X ! Y is proper and flat. Then:

(i) We have f + (F) = f ⇤ (F ⌦ ! X/Y ).
(ii) the functor

F 7 ! f ! (F) := f ⇤ (F ⌦ ! X/Y
) is a right adjoint for the functor f ⇤ .

Contents 5.1 Introduction

Main results

Let X be a proper and smooth scheme over an algebraically closed field. The goal of the present text is to show the existence of the moduli of rank n étale p-adic lisse sheaves on X, study its geometry and its corresponding deformation theory. More precisely, let k denote a non-archimedean field extension of Q p . We will construct a functor LocSys `,n (X) : Afd op k ! S, where Afd k denotes the category of k-affinoid spaces and S the 1-category of 1-groupoids, given on objects by the formula

A 2 Afd op k 7 ! LocSys `,n (X)(A) 2 S
where LocSys `,n (X)(A) denotes the groupoid of conjugation classes of continuous morphisms

⇢ : ⇡ ét 1 (X) ! GL n (A)
where we endow GL n (A) with the topology induced by the non-archimedean topology on A 2 Afd op k . Our first main result is the following: (X) as a smooth atlas of LocSys `,n (X). Moreover, LocSys framed `,n (X) admits a canonical action of the k-analytic group GL an n and LocSys `,n (X) can be realized as the stack quotient of LocSys framed `,n (X) by the GL an n -action.

We can construct LocSys framed `,n (X) explicitly via its functor of points. Explicitly, LocSys framed `,n (X) represents the functor Afd op k ! Set given on objects by the formula

A 2 Afd op k 7 ! Hom cont ⇡ ét 1 (X), GL n (A) 2 Set. (5.1.1.1)
Showing that the functor given by formula (5.1.1.1) is representably by a k-analytic space LocSys framed `,n (X) 2 An k will occupy most of §2. Our proof follows the scheme of proof of the analogous result for smooth and proper schemes over the field of complex numbers C. However, our argument is considerably more involved as in general the topologies on ⇡ ét 1 (X) and GL n (A) are of different natures. More precisely, the former admits a profinite topology whereas the latter group admits a ind-pro-topology, where the pro-structure comes from the choice of a formal model A 0 for A and the ind-structure by the existence of an isomorphism

A 0 ⌦ k k ⇠ = A
125 of topological algebras. Moreover, thanks to the formula (5.1.1.1) it is clear that LocSys framed `,n (X) admits a canonical action of the k-analytic general linear group GL an n , given by conjugation. The rest of §2 is devoted to present the theory of k-analytic stacks and to show that LocSys `,n (X) can be identified with the k-analytic stack obtained by "quotientening" LocSys framed `,n (X) by GL an n . We observe that k-analytic stacks are the k-analytic analogues of Artin stacks in algebraic geometry.

We show moreover that our definition of LocSys `,n (X) is correct. More precisely we show:

Proposition 5.1.1.2. Let A 2 Afd op k . Then the groupoid LocSys `,n (X)(A) 2 S
can be identified with the groupoid of rank n pro-étale A-local systems on X.

We then proceed to study the deformation theory of LocSys `,n (X). We prove more precisely that LocSys `,n (X) admits a canonical derived enhancement. By derived enhancement we mean that there exists a derived k-analytic stack, following Porta and Yu Yue approach to derived k-analytic geometry [PY17a], whose 0-truncation is naturally equivalent to LocSys `,n (X).

In order to construct such derived structure on LocSys `,n (X) we need to first extend its definition to derived coefficients, i.e. we need to extend LocSys `,n (X) to a functor defined on the 1-category of derived k-analytic spaces dAfd k , such that when restricted to the full subcategory of discrete objects Afd k ,! dAfd k we recover the k-analytic stack LocSys `,n (X). In order to provide a correct definition of a derived enhancement of LocSys `,n (X) we employ the language of enriched 1-categories. Namely, given Z 2 dAfd k a derived k-affinoid a continuous representation

⇢ : Sh ét (X) ! BGL n (Z) ,
where Sh ét (X) denotes the étale homotopy type of X and : dAfd op k ! CAlg k the derived global sections functor, corresponds to an object in the 1-category of functors

Fun Cat1(Ind(Pro(S))) Sh ét (X), Perf( (Z)) , (5.1.1.2)
where we interpret the 1-category of perfect complexes Perf( (Z)) as enriched over Ind(Pro(S)) in a suitable sense. We will explore these constructions in both §4 and §5. More precisely, in §4 we treat the case of continuous representations ⇢ : Sh ét (X) ! Perf(A) where A is a derived k -adic algebra. Studying derived k -adic continuous representations of Sh ét (X) will prove useful in order to show that the 1-category Fun Cat1(Ind(Pro(S))) Sh ét (X), Perf( (Z)) 2 Cat 1 satisfies many pleasant conditions. We deal with this in §5, where we prove new results concerning the lifting of continuous representations ⇢ : Sh ét (X) ! BEnd( (Z)) to a continuous representation ⇢ : Sh ét (X) ! BEnd(A) where A is a derived k -adic algebra such that Spf(A) is a formal model for Z 2 dAfd k . This is possible thanks to results concerning the existence of formal models for derived k-analytic spaces, proved in [Ant18b].

We will then show that when we restrict ourselves to the full subcategories of (5.1.1.2) spanned by rank n free modules we get the desired derived enhancement of LocSys `,n (X). With this knowledge at our disposal we are able to show the following important result: Theorem 5.1.1.3. The k-analytic stack LocSys `,n (X) : Afd op k ! S admits a derived enhancement, which we denote RLocSys `,n (X). Moreover, the derived moduli stack RLocSys `,n (X) admits a global analytic cotangent complex. Given a Z 2 dAfd k -point of RLocSys `,n (X)

⇢ : Z ! RLocSys `,n (X)
the analytic cotangent complex of RLocSys `,n (X) is canonically equivalent to

L an RLocSys `,n (X),⇢ ' C ⇤ ét X, Ad(⇢) _ [ 1] 2 Mod (Z) ,
where C ⇤ ét X, Ad(⇢) _ denotes the complex of étale cochains on the étale site of X with values in the derived local system

Ad(⇢) := ⇢ ⌦ ⇢ _ .
Using the main theorem [PY17a, Theorem 7.1] we are thus able to show the following second main result:

Theorem 5.1.1.4. The functor RLocSys `,n (X) : dAfd op k ! S is representable by a derived k-analytic stack whose 0-truncation agrees canonically with LocSys `,n (X).

Notations and Conventions

We shall denote k a non-archimedean field equipped with a non-trivial valuation, k its ring of integers and sometimes we will use the letter t 2 k to denote a uniformizer for k. We denote An k the category of strict k-analytic spaces and Afd k the full subcategory spanned by strict k-affinoid spaces and we adopt the convention that whenever we mention k-affinoid or k-analytic space we mean strict k-affinoid and strict k-analytic space, respectively. We denote fSch k the category of quasi-separated formal schemes over the formal spectrum Spf(k ), where we consider k equipped with its canonical topology induced by the valuation on k. In order to make clear that we consider formal schemes over Spf(k ), we shall often employ the terminology k -adic scheme to refer to formal scheme over Spf k .

Let n 1, we shall make use of the following notations:

A n k := Spec k[T 1 , . . . , T m ], A n k := Spf(k hT 1 , . . . , T m i) and A n k := A n k an , B n k := Sp(khT 1 , . . . T m i)
, where ( ) an denotes the usual analytification functor ( ) an : Sch k ! An k , see [Ber93a]. We denote by GL an n the analytification of the usual general linear group scheme over k, which associates to every k-affinoid algebra A 2 Afd k the general linear group GL n (A) with A-coefficients.

In this thesis we extensively use the language of 1-categories. Most of the times, we reason model independently, however whenever needed we prove 1-categorical results using the theory of quasi-categories and we follows closely the notations in [Lur09b]. We use caligraphic letters C, D to denote 1-categories. We denote Cat 1 the 1-category of (small) 1-categories. We will denote by S the 1-category of spaces, S fc the 1-category of finite constructible space, see [Lur09a, §3.1]. Let C be an 1-category, we denote by Ind(C) and Pro(C) the corresponding 1-categories of ind-objects and pro-objects on C, respectively. When C = S fc , the 1-category

Pro(S fc

) is referred as the 1-category of profinite spaces. Let R be a derived commutative ring. We will denote by CAlg R the 1-category of derived k-algebras. The latter can be realized as the associated 1-category to the usual model category of simplicial R-algebras.

We shall denote by CAlg ad the 1-category of derived adic algebras, introduced in [Lur16, §8.1]. Whenever R admits a non-trivial adic topology, we denote CAlg ad R := CAlg ad R/ the 1-category of derived adic Ralgebras, i.e. derived R-algebras equipped with an adic topology compatible with the adic topology on R together with continuous morphisms between these.

Let R be a field. We shall denote by CAlg sm R the 1-category of small augmented derived R-algebras. When R = k we denote by AnRing sm k the 1-category of small augmented derived k-analytic rings over k, which is naturally equivalent to CAlg sm k , see [Por15a, §8.2]. Let R be a discrete ring. We denote by CAlg R the 1-category of ordinary commutative rings over R. When R admits an adic topology we shall denote CAlg ad,R ✓ CAlg ad R the full subcategory spanned by discrete derived adic R-algebras. Let R denote a derived ring. We denote Mod R the derived 1-category of R-modules and Coh + (X) ✓ Mod R the full subcategory spanned by those almost perfect R-modules. We need sometimes to enlarge the starting Grothendieck universe, and we often do not make explicit such it procedure. Fortunately, this is innocuous for us. Given Z 2 dAfd k a derived k-affinoid space and M 2 Coh + (Z)

an almost perfect sheaf on Z its mapping space of endomorphisms End(M ) 2 S admits a natural enrichment over the 1-category Ind(Pro(S)). We shall denote such enrichement by End(M ). We will employ the same notation whenever M 2 Coh + (A) where A 2 CAlg ad k . Namely, for such M 2 Coh + (A) we denote End(M ) the E 1 -monoid like object on the 1-category Pro(S).

We will denote by dA↵ k , ⌧ ét , P sm the algebraic geometric context and we denote by dSt dA↵ k , ⌧ ét , P sm the 1-category of derived geometric stacks with respect to dA↵ k , ⌧ ét , P sm . Similary, whenever k denotes either the field C of complex numbers or a non-archimedean field we will denote by dA↵ k , ⌧ ét , P sm the analytic geometric context and correspondingly dSt dAn k , ⌧ ét , P sm the 1-category of derived geometric stacks with respect to the analytic geometric context.

Acknowledgments

I would like to express my deep gratitude to my advisor B. Toen for all his advice and by sharing many ideas and suggestions concerning the contents of this text. I am also in great debt to M. Porta from which I learned derived non-archimedean geometry. I am thankful to Mauro Porta also for so many useful mathematical suggestions along these last years. I would also like to thank to M. Robalo, A. Vezzani, B. Hennion, D. Gapner, M. d'Addezio, V. Daruvar, C. Simpson, V. Melani, P. Scholze, B. Bhatt, B. Conrad, B. Stroh, and J. Tapia for useful commentaries and suggestions during the elaboration of the text. I would also like to express my gratitude to the establishment Université Paris 13 and Université de Strasbourg, where I could discuss and share many of the ideas presented in this text.

Representability of the space of morphisms

Let G be a profinite group topologically of finite generation. One can consider the functor

LocSys framed `,n (G) : Afd op k ! Set
given on objects by the formula

A 2 Afd k 7 ! Hom cont G, GL n (A) 2 Set,
where Hom cont G, GL n (A) denotes the set of continuous group homomorphisms

G ! GL n (A),
where we consider GL n (A) as a topological group via the induced topology on A 2 Afd k . We will prove that

LocSys framed `,n (
G) is representable by a k-analytic space, i.e.

LocSys framed `,n (X) 2 An k .
The proof of representability is established first when G is a free profinite group. This is the main result of the section. The case where G is a more general topologically finitely generated profinite group follows directly from the case of topologially free profinite groups.

Our main motivation to study LocSys framed `,n (G) follows from the fact that it forms a smooth atlas of the moduli of continuous representations of G, which we shall designate the latter by LocSys `,n (G). One can show that LocSys `,n (G) is equivalent to the "stack-quotient" of LocSys framed `,n (G) by its natural action of the k-analytic general linear group GL an n under conjugation. Furthermore, the representability of LocSys framed `,n (G) entails the representability of LocSys `,n (G) as a geometric stack with respect to the k-analytic context. We shall prove this latter assertion in §2.3 and review the main basic notions concerning k-analytic stacks.

Preliminaries

This § reviews the basic notions of k-analytic geometry that we will use more often. Definition 5.2.1.1. Let n 1 be an integer. The Tate k-algebra on n generators with radius (r 1 , . . . , r n ) is defined as khr 1 1 T 1 , . . . , r Definition 5.2.1.3. Let A be a k-affinoid algebra we say that A is strict k-affinoid if we can choose such a presentation for A with the r i = 1, for each i. We denote by Afd op k the category of strict k-affinoid algebras together with continuous k-algebra homomorphisms between them.

Remark 5.2.1.4. The k-algebra khr 1 1 T 1 , . . . , r 1 n T n i admits a canonical k-Banach structure induced by the usual Gauss norm. Moreover, any finitely generated ideal I ⇢ khr 1 1 T 1 , . . . , r 1 n T n i is closed which implies that any k-affinoid algebra A admits a k-Banach structure, depending on the choice of a presentation of A. Nonetheless it is possible to show that any two such k-Banach structures on A are equivalent and therefore the latter inherits a canonical topology, induced from the one on khr 1 1 T 1 , . . . , r 1 n T n i given by the Gauss norm. Strict k-affinoid algebras correspond to the affine objects in (rigid) k-analytic geometry. Therefore, we define the category of k-affinoid spaces as In Berkovich's non-archimedean geometry it is possible to define the notion of relative interior, which is very useful in practice. Let : A ! A 0 denote a bounded morphism of k-affinoid algebras. The relative interior of , denote Int(M(A 0 )/M(A)) is by definition the set of points,

Int M(A 0 )/M(A) := {x 0 2 M(A 0 )|A 0 ! H(x 0
) is inner with respect to A}, where inner with respect to A means that there exist a continuous surjective map Ahr 1 1 T 1 , . . . , r 1 n T n i ! A 0 of k-affinoid algebras which induces a norm on A 0 equivalent to its original one and such that, for each i, we have

|T i | x 0 < r i .
Definition 5.2.1.8. One can glue k-affinoid spaces, as in algebraic geometry. A k-analytic space is defined as a locally ringed space which locally is equivalent to a k-affinoid space. We denote by An k the category of k-analytic spaces and morphisms between these.

One then is able to globalize most of the previous notions, in particular it is possible to give a global definition of the relative interior of a morphism between k-analytic spaces. We refer the reader to [Ber93a], [Con08a] and [Bos05] for a more detailed exposition on rigid geometry, from different points of view.

Hom spaces

Let G denote a profinite group of topological finite type which we fix throughout this §. Consider the functor

LocSys framed `,n : Afd op k ! Set
given informally on objects by the formula

A 7 ! Hom cont G, GL n (A) ,
where Hom cont denotes the set of morphisms in the category of continuous groups, and we consider GL n (A) with the topology induced by the topology of A viewed as a k-affinoid algebra. Our goal in this section is to show that

LocSys framed `,n (G) 2 Fun Afd op k , Set
is representable by a k-analytic space. Let A 2 Afd op k be a (strictly) k-affinoid algebra. Notation 5.2.2.1. We will typically denote by A 0 a formal model for A, i.e., a (p-adic complete) k -algebra of topological finite presentation such that we have an isomorphism

A 0 ⌦ k k ' A.
Remark 5.2.2.2. By choice of A 0 , we conclude that A 0 can be identified with an open subring of A. For this reason, the topology of A can be thought as an ind-pro topology, in which the pro-structure comes from the fact that formal models are p-adic complete and the ind-structure arises after localizing at p.

Id + p k+1 • M n (A 0 ) E GL n (A 0 ).
These form a basis of normal open subgroups for the topology on GL n (A 0 ) induced by A 0 . We have moreover canonical isomorphisms

GL n (A 0 )/ Id + p k M n (A 0 ) ' GL n (A 0 /p k A 0 ). We have thus a canonical isomorphism GL n (A 0 ) ⇠ = lim k 1 GL n (A 0 )/ Id + p k M n (A 0 ) .
Thus it is p-adically complete. The same reasoning holds for the topological group Id + p k • M n (A 0 ), for each k 1. More concretely, we have isomorphisms

Id + p k • M n (A 0 ) ⇠ = lim m 1 Id + p k • M n (A 0 )/(Id + p k+m • M n (A 0 ) .
Notation 5.2.2.4. We denote by b F r a fixed free profinite group of rank r. It can be explicitly realized as the profinite completion of a free group on r generators, F r . The latter can be realized as a dense full subgroup of b F r . We will thus fix throughout the text a continuous dense group inclusion homomorphism F r ! b F r and a set of generators e 1 , . . . , e r 2 F r which become topological generators of the profinite group b F r .

Remark 5.2.2.5. Let FinGrp denote the category of finite groups. The category of profinite group corresponds to its pro-completion, Pro(FinGrp). For each r 1, the groups b F r 2 Pro(FinGrp) satisfy the universal property given by the formula

Hom Pro(FinGrp) ( b F r , G) ⇠ = G r ,
⇢ : b F r ! GL n (A) such that M 1 := ⇢(e 1 ), . . . , M r := ⇢(e r ). We denote by M 1 , . . . M r := Y i M nj i i 2 GL n (A),
whenever the right hand side is well defined, (which is always the case when the product on the left hand side is indexed by a finite set). Definition 5.2.2.9. Let U 2 J r and fix 1 , . . . , l 2 U \ b F r a finite number of topological generators for U . We define the functor

LocSys framed `,n (U, 1 , . . . , l ) : Afd k ! Set,
given on objects by the formula,

A 7 ! LocSys framed `,n (U, 1 , . . . , l )(A) := {(M 1 , . . . , M r ) 2 GL n (A) r : for each i 2 [1, l], | i (M 1 , . . . , M r ) Id|  |p|}.
(U, 1 , . . . , l )(A 0 ) = LocSys framed `,n (U, ⌧ 1 , . . . , ⌧ s )(A 0 ).
In order to see this, it suffices to note that, for each n 1, the mod p n reduction of

(M 1 , . . . , M r ) 2 LocSys framed `,n (U, 1 , . . . , l )(A 0 /⇡ n A 0 )
corresponds to a group homomorphism

:= b F r /U ! GL n (A 0 /⇡A 0 ).
As group homomorphisms are independent of the choice of presentation for the set

LocSys framed `,n (U, 1 , . . . , l )(A 0 /⇡ n ) 2 Set
does not depend on such choices, either. As A 0 is ⇡-adic complete" passing to inverse limits we deduce that

LocSys framed `,n (U, 1 , . . . , l )(A 0 )
is independent of the choice of topological generators for U , as desired. (U )(A 0 ) 2 Set.

Lemma 5.2.2.12. Let A 2 Afd op k be an k-affinoid algebra and A 0 an k -formal model for A. Then there is a bijection

Hom cont b F r , GL n (A 0 ) ⇠ = colim U 2Jr LocSys framed `,n U (A 0 ),
of sets, for each r 1.

Proof. Since

Hom cont b F r , GL n (A 0 )) denotes the set of continuous group homomorphisms in the category of pro-discrete groups we have a bijection

Hom cont b F r , GL n (A 0 ) ⇠ = lim k colim 2Jr Hom Grp , GL n (A 0 /p k+1 A 0 ) .
of sets. It therefore suffices to show that we have a bijection,

lim k colim U 2Jr Hom Grp U , GL n (A 0 /p k+1 A 0 ) ⇠ = colim U 2Jr LocSys framed `,n U (A 0 ),
where U denotes the finite group b F r /U . We assert that there exists a canonical morphism,

: lim k colim U 2Jr Hom Grp U , GL n (A 0 /p k+1 A 0 ) ! colim U 2Jr LocSys framed `,n U (A 0 ).
In order to prove this assertion we observe that a group morphism

⇢ k : U k ! GL n (A 0 /p k+1 A 0 ),
with U k 2 J r , is determined by the image of the r generators of U k which correspond to r matrices in

GL n (A 0 /p k+1 A 0 )
. Therefore given such a system of compatible group homomorphisms {⇢ k } k one can associate an r-vector (M 1 , . . . , M r ) 2 GL n (A 0 ) r such that its mod p reduction satisfies i (M 1 , . . . , M r ) = Id, where 1 , . . . , l 2 U \ F r denotes a choice of a finite set of topological generators for U 1 . Thus

(M 1 , . . . , M r ) 2 X U1 (A 0 ).
This shows the existence of the desired map. We now construct maps

U : LocSys framed `,n U (A 0 ) ! lim k colim U 0 2Jr Hom Grp , GL n (A 0 /p k+1 A 0 ) ,
for each U 2 J r , such that when we assemble these together we obtain the desired inverse for . In order to construct U , we start by fixing topological generators

1 , . . . , l 2 U \ F r for U . Let (M 1 , . . . , M r ) 2 LocSys framed `,n
U (A 0 ). As we have seen these matrices define a continuous group homomorphism b F r ! GL n (A 0 /pA 0 ). Thanks to Theorem 5.2.2.13 below the matrices 1 (M 1 , . . . , M r ), . . . , l (M 1 , . . . , M r ) 2 Id + pM n (A 0 ) determine a continuous group homomorphism

⇢ 1 : U ! Id + pM n (A 0 ).

Then the inverse image

U 0 2 := ⇢ 1 1 Id + p 2 M n (A 0 )
is an open normal subgroup of U of finite index. As U itself is an open subgroup of b F r of finite index we conclude that U 0 2 is also a finite index subgroup of b F r . As open normal subgroup of finite index in b F r define a final family for b F r we conclude that there exists U 2 2 J r such that ⇢ 1 (U 2 ) is a subgroup of Id + p 2 M n (A 0 ). Consequently, the matrices (M 1 , . . . , M r ) 2 X U (A 0 ) define a group homomorphism

⇢ 2 : b F r /U 2 ! GL n (A 0 /p 2 A 0 ).
By iterating the process we obtain a sequence of continuous group homomorphisms

{⇢ i : b F r /U i ! GL n (A 0 /p i A 0 )} i 2 lim i colim U 2Jr Hom grp ( U , GL n (A 0 /p i A 0 )).
Assembling these together we obtain a continuous group homomorphism ⇢ 2 Hom cont ( b F r , GL n (A 0 ). It follows easily by our construction that,

colim U 2Jr ( U ) : colim U 2Jr LocSys framed `,n U (A 0 ) ! Hom cont b F r , GL n (A 0 ) ,
is the inverse map of , as desired.

Lemma 5.2.2.13 (Burnside problem for topologically nilpotent p-groups). Let A 2 Afd op k and A 0 be a formal model for A. For each k 1, we have a natural bijection

Hom cont b F r , Id + p k+1 M n (A 0 ) = Id + p k+1 M n (A 0 ).
Proof. Noticing that the quotient groups,

Id + p k M n (A 0 )/(Id + p k+m+1 M n (A 0 ) ,
are torsion, i.e., every element has finite order and we conclude that

Hom cont b Z, Id + p k+1 M n (A 0 ) = Id + p k+1 M n (A 0 ),
where b Z denotes the profinite completion of Z. This finishes the proof when r = 1. The same holds for general b F r , i.e., we have a canonical equivalence,

Hom cont b F r , Id + p k+1 M n (A 0 ) = Id + p k+1 M n (A 0 ) r .
In order to prove this last assertion it suffices to show that any finitely generated subgroup of the quotient

Id + p k M n (A0)/(Id + p k+m+1 M n (A 0 ) ,
for some positive integer m 1, is finite (i.e. the Burnside problem admits an affirmative answer in this particular case). In order to justify the given assertion we fix G a finitely generated subgroup of

Id + p k M n (A 0 )/(Id + p k+l+1 M n (A 0 ) .
By assumption it is generated by matrices of the form Id + p k+1 N 1 , . . . , Id + p k+1 N s , for some s 1. Therefore a general element of G can be written as,

Id + p k+1 (n 1,1 N 1 + • • • + n 1,s N s ) + • • • + p k+l 1 (n a 1,1 N a 1,1 + • • • + n a 1,s l 1 N a 1,s l 1 )
,

where the N i,j , for i, j 2 [1, a 1] ⇥ [1, s a 1
], denote products of the N i having a 1 multiplicative terms, where a denotes the least integer such that k ⇥ (a + 1) l. By the Pigeonhole principle there are only finite number of such choices for the integers n i,j for (i, j)

2 [1, l 1] ⇥ [1, s l 1
] and the result follows.

Proposition 5.2.2.14. Let A 2 Afd op k be an k-affinoid algebra then we have a natural bijection,

Hom cont b F r , GL n (A) ' colim U 2Jr, 1 ,..., l generators LocSys framed `,n
U, 1 , . . . , l (A).

Proof. Let ⇢ : b F r ! GL n (A) be a continuous homomorphism of topological groups and let e 1 , . . . , e r be the fixed topological generators of b F r . Let

M i := ⇢(e i ) 2 GL n (A)
for each

1  i  r. The group Id + p • M n (A 0 ) is open in GL n (A 0 )
and the latter open in GL n (A). We thus deduce that the inverse image

U := ⇢ 1 Id + pM n (A 0 )
is an open subgroup of b F r and it has thus finite index in b F r , moreover as J r is a final family for b F r one can suppose without loss of generality, up to shrinking U , that U 2 J r and thus normal in b F r . Choosing a finite set of topological generators for ⇢ 1 Id + ⇡ • M n (A 0 ) we deduce that the (M 1 , . . . , M r ) satisfy the inequalities, in GL n (A), associated to such generators, therefore

(M 1 , . . . , M r ) 2 colim U 2Jr,generators 1 ,..., l LocSys framed `,n U, 1 , . . . , l (A)
which proves the direct inclusion. We conclude that the association

⇢ 2 Hom cont ( b F r , GL n (A)) 7 ! (⇢(e 1 ), . . . , ⇢(e r )) 2 colim U 2Jr, 1 ,..., l generators LocSys framed `,n U, 1 , . . . , l (A),
defines a well defined map of sets. Let us prove that we have a well defined inverse map. We consider

(M 1 , . . . , M r ) 2 GL n (A) r such that i (M 1 , . . . , M r ) 2 Id + ⇡ • M n (A 0 ) for a finite family { i } i2[1,l] ,
all lying in the dense subgroup

F r ⇢ b F r , of
⇢ : U ' F pf l ! Id + pM n (A 0 ).
Therefore, we have the following diagram in the category of topological groups,

F r b F r U GL n (A) GL n (A) Id + pM n (A 0 ). (M1,...,Mr) =
We want to show that we can fill the above diagram with a continuous morphisms b F r ! GL n (A) making the whole diagram commutative. Since U is of finite index in b F r we can choose elements

g 1 , . . . g m 2 F r ⇢ b F r
such that these form a (faithful) system of representatives for the finite group b F r /U . For i 2 [1, m] write

g i := Y ji e nj i ji , 134
where this product is finite and unique by the assumption that the g i 2 F r . Every element of h 2 b F r can be written as h = g i , for some g i as above and 2 U . Let us then define

⇢(h) := Y ji M nj i ji ⇢( ) 2 GL n (A).
We are left to verify that the association

h 2 b F r 7 ! ⇢(h) 2 GL n (A)
gives a well defined continuous group homomorphism. Let

g := Y s e ns s 2 F r ⇢ b F r
and 0 2 U such that g 0 = h = g i . We first prove that

⇢(h) = Y s M ns s ⇢( 0 ).
Suppose that , 0 2 U \ F r , then it follows that h 2 F r . The result then follows in this case, since we have fixed a group homomorphism (M 1 , . . . , M r ) : F r ! GL n (A), which is necessarily continuous. Suppose then that it is not the case that

, 0 2 U \ F r .
Let ( n ) n and ( n 0 ) n 0 be sequences of elements in U \ F r converging to and 0 , respectively. We observe that this is possible since F r is dense in b F r and U \ F r is a free (discrete) group whose profinite completion is canonically equivalent to U , thus dense in U . For this reason, we obtain that

g 1 g i = 0
and we get moreover that g 1 g i n converges to 0 . Thus the elements

( Y s M n s 1 s 1 )( Y ji M nj i ji )⇢( m ) 2 GL n (A) ,
where our notations are clear from the context, converge to ⇢( 0) by continuity of ⇢. They also converge to the element

( Y s M n s 1 s 1 )( Y ji M nj i ji ) 2 GL n (A)
by continuity of the group multiplication on GL n (A). Since the topology on A comes from a norm on A, making the latter a Banach k-algebra we conclude that A is Hausdorff and so it is GL n (A). This implies that converging sequences in GL n (A) admit a unique limit. We conclude therefore that,

⇢( 0 ) = ( Y s M n s 1 s 1 )( Y ji M nj i ji )⇢( ),
thus giving the desired equality,

( Y s M ns s )⇢( 0 ) = ( Y ji M nj i ji )⇢( ),
proving that ⇢ : b F r ! GL n (A) is a well defined map. We wish to show that it is a continuous group homomorphism. Our definitions make clear that to check multiplicativity of ⇢ it suffices to show that for every g 2 F r and sigma 2 U we have, ⇢(g g 1 ) = ⇢(g)⇢( )⇢(g 1 ).

Pick again a converging sequence ( n ) n , in F r \ U , such that n converges to . Then for each n we have,

⇢(g n g 1 ) = ⇢(g)⇢( n )⇢(g 1 ),
and by passing to the limit, we obtain the desired equality. We are reduced to show that ⇢ is continuous. Let V be an open subset of GL n (A). The intersection V \ (Id + pM n (A 0 ) is open in GL n (A). Thus, n and (M 1 , . . . , M r ) the r-vector whose matrix components correspond to identity morphism of GL 0 n ). Let c i 2 |k ⇥ | be a decreasing sequence of real numbers converging to 0, there is a natural isomorphism

⇢ 1 (V \ (Id + pM n (A 0 ) = ⇢ 1 (V \ (Id + pM n (A 0 )) is open in U . Therefore, the quotient U/(V \ (Id + pM n (A 0 )) is discrete, since U is of finite index in b F r . We conclude that, U/⇢ 1 (V \ (Id + pM n (A 0 )) ! b F r /⇢ 1 (V ) exhibits the quotient U/⇢ 1 (V \ (Id + pM n (A 0 )) as a subgroup of finite index in b F r /⇢ 1 (V ).
(GL 0 n ) r ' colim i (GL 0 n ) r ci ,
where (GL 0 n ) r ci denotes a copy of (GL 0 n ) r indexed by c i , and the inclusion morphisms in the corresponding diagram sends (GL 0 n ) r ci to the closed disk of radius c 1 i inside of (GL 0 n ) r ci+1 . Henceforth we have canonical isomorphisms

LocSys framed `,n U, 1 , . . . , l ⇠ = colim i LocSys framed `,n U, 1 , . . . , l 0 ci ,
which is a union of k-affinoid subdomains where the image of an element in the filtered diagram lies in the interior, in Berkovich's sense, of the successive one. We thus conclude that LocSys framed `,n U, 1 , . . . , l is itself representable by an k-analytic space. Proof. We start by remarking that if U 0 ⇢ U is an inclusion of subgroups lying in the family J r then they induce an inclusion

LocSys framed `,n U, 1 , . . . , l ,! LocSys framed `,n U 0 , ⌧ 1 , . . . , ⌧ s . We employ the notation i (M 1 , . . . , M r ) 2 Id + pM n (A 0 )
where the i denote a choice of generators for U , lying in the dense subgroup U \ F r . It follows that we have necessarily are nice enough whenever U denotes a sufficiently large (finite) index subgroup of b F r . Fix U 2 J r and let 1 , . . . , l be a finite set of generators for U . Given a k-affinoid algebra A and

⌧ 0 j (M 1 , . . . , M r ) 2 Id + pM n (A 0 ) for a choice of generators for U 0 , lying in U 0 \ F r , denoted ⌧ 1 , . . . ⌧ s .
(M 1 , . . . , M r ) 2 LocSys framed `,n U, 1 , . . . , l (A)
we can write i (M 1 , . . . , M r ) = Id + p • N i , for suitable matrices N i 2 M n (A0), for each i. Moreover the r-tuple (M 1 , . . . , M r ) define a continuous group homomorphism

⇢ : b F r ! GL n (A)
. By Theorem 5.2.2.13, quotients of the pro-p-group Id + pM n (A 0 ) are of p-torsion. Let U 0 2 J r such that

U 0 ⇢ ⇢ 1 (Id + p 2 M n (A 0 )).
Given ⌧ 1 , . . . , ⌧ s generators for U 0 as above we have

|⌧ i (M 1 , . . . , M r ) Id|  |p 2 | < |p|, for each i 2 [1, s]. This implies that (M 1 , . . . , M r ) 2 Int LocSys framed `,n U 0 , ⌧ 1 , . . . , ⌧ s /(GL an n ) r , For each U 2 J r , LocSys framed `,n
U, 1 , . . . , l is a k-analytic subdomain of (GL an n ) r , and therefore

Int LocSys framed `,n U, 1 , . . . , l /(GL an n ) r ,! (GL an n ) r
is an open subset of (GL an n ) r , [Con08a, Exercise 4.5.3]. Moreover, the functor

LocSys framed `,n ( b F r ) ,! : Afd op k ! Set
is a subfunctor of (GL an n ) r , which follows readily from the definitions. We can therefore (canonically) associate to LocSys framed `,n ( b F r ) a topological subspace

X := [ 2Jr LocSys framed `,n U, 1 , . . . , l ⇠ = [ 2Jr Int LocSys framed `,n (U, 1 , . . . , l )/(GL an n ) r ,
Therefore, the topological space X corresponds to an open subspace of the underlying topological space of (GL an n ) r . Consequently, the former is necessarily an Hausdorff space. We will construct a canonical k-analytic structure on it and show that such k-analytic space represents the functor LocSys framed `,n

( b F r ). As each LocSys framed `,n U, 1 , . . . , l 2 An k
is a k-analytic space we can take the maximal atlas and quasi-net on it consisting of k-affinoid subdomains of LocSys framed `,n U, 1 , . . . , l , which we denote by T U, 1 ,..., l . As X can be realized as a filtered union of the LocSys framed `,n U, 1 , . . . , l we conclude that the union of the quasi-nets T U, 1 ,..., l induces a quasi-net T on LocSys framed `,n ( b F r ). In order to prove this we shall show that given a point x 2 X we need to be able to find a finite collection V 1 , . . . , V n of compact Hausdorff subsets of X such that x 2 We are reduced to show that the union

T i V i and moreover V 1 [ • • • [ V n is an open neighborhood of x inside X .
V 1 [ • • • [ V n is open in LocSys framed `,n ( b F r ). By shrinking the V i of larger index in b F r , if necessary, we can assume that the union V 1 [ • • • [ V n lies
Int(LocSys framed `,n U, 1 , . . . , l /(GL an n ) r . The latter is also open in (GL an n ) r , consequently also the union V 1 [ • • • [ V n is open in (GL n ) r . We conclude that there exists an open subset W of (GL an n ) r such that V 1 [ • • • [ V n = LocSys framed `,n U, 1 , . . . , l \ U = Int LocSys framed `,n U, 1 , . . . , l /(GL an n ) r \ W and therefore V 1 [ • • • [ V n is itself open in (GL an n ) r . We conclude thus that V 1 [ • • • [ V n is also open in LocSys framed `,n
( b F r ), as desired. Clearly, T induces quasi-nets on the intersections W \ W 0 for any W, W 0 2 T as we can always choose a sufficiently large finite quotient of b F r such that W, W 0 ⇢ LocSys framed `,n U, 1 , . . . , l . The fact that the union of the maximal atlas on each LocSys framed `,n U, 1 , . . . , l gives an atlas on X , with respect to T, is also clear from the definitions. We conclude that the topological space X is endowed with a natural structure of k-analytic space (in fact, a k-analytic subdomain of (GL an n ) r ). We shall show that the k-analytic space X represents the functor

LocSys framed `,n ( b F r ) : Afd op k ! Set.
As k-affinoid spaces are compact we conclude that any map Sp B A ! X factor through some LocSys framed `,n U, 1 , . . . , l as their union equals the union of the respective relative interiors. As Sp B A is quasi-compact, it follows from Theorem 5.2.2.14, that the functor of points, thanks to associated to X is canonically equivalent to LocSys framed `,n ( b F r ) and the result follows.

Corollary 5.2.2.19. Let G be a profinite group topologically of finite type then the functor

LocSys framed `,n (G) : Afd op k ! Set,
given on objects by the formula

A 2 Afd op k ! Hom cont G, GL n (A) 2 Set,
is representable by a k-analytic space.

Proof. Let us fix a continuous surjection of profinite groups

q : b F r ! G,
for some integer r  1. Let H denote the kernel of q. Thanks to Theorem 5.2.2.18 we know that LocSys framed `,n ( b F r ) is representable by a k-analytic stack. We have an inclusion at the level of functor of points

q ⇤ : LocSys framed `,n (G) ! LocSys framed `,n ( b F r )
induced by precomposing continuous homomorphisms ⇢ : G ! GL n (A) with q. We show that the morphism q ⇤ is representable and a closed immersion. Let Sp B A be a k-affinoid space and suppose given a morphism of k-analytic spaces,

⇢ : Sp B A ! LocSys framed `,n ( b F r ),
which corresponds to a continuous representation ⇢ : G ! GL n (A). We want to compute the fiber product

Sp B A ⇥ LocSys framed `,n ( b Fr) LocSys framed `,n (G).
Since Sp B A is quasi-compact and we have an isomorphism at the underlying topological spaces,

LocSys framed `,n ( b F r ) ⇠ = colim U 2Jr,generators 1 ,..., l Int X U, 1 ,..., l /(GL n ) r ,
we conclude that

⇢ : Sp A ! LocSys framed `,n ( b F r )
factors through a k-analytic subspace of the form LocSys framed `,n U, 1 , . . . , l , for suitable such U 2 J r and 1 , . . . , l . By applying again the same reasoning we can assume further that

⇢ : Sp B A ! LocSys framed `,n ( b F r )
factors through some LocSys framed `,n U, 1 , . . . , l 0 as in the proof of Theorem 5.2.2.17. The latter is k-affinoid, say X 0 U, 1 ,..., l ⇠ = Sp B B in the category Afd k , for some k-affinoid algebra B. Let X 0 G,U, 1 ,..., l denote the fiber product,

LocSys framed `,n G, U, 1 , . . . , l 0 LocSys framed `,n (G) LocSys framed `,n U, 1 , . . . , l 0 LocSys framed `,n ( b F r )
.

By construction, the set

X 0 G,U, 1 ,..., l (A) 2 Set

Geometric contexts and geometric stacks

Our next goal is to give an overview of the general framework that allow us to define the notion of a geometric stack. Our motivation comes from the need to define the moduli stack of continuous representations of a profinite group G (of topological finite presentation) as a non-archimedean geometric stack. This latter object should be obtained by taking the quotient of LocSys framed `,n (G) by the conjugation action of GL an n on LocSys framed `,n (G). We will review these notions and show that such we are able to construct such a quotient via a formal procedure. (ii) The class P is closed under equivalences, compositions and pullbacks.

(iii) Every ⌧ -covering consists of moprhisms in P.

(iv) For any morphism f : X ! Y in C, if there exists a ⌧ -covering {U i ! X} such that each composition U i ! Y belongs to P then f belongs to P. Given a geometric context (C, ⌧, P) it is possible to form an 1-category of geometric stacks Geom(C, ⌧, P) via an inductive definition as follows:

Definition 5.2.3.3. A morphism in F ! G in Shv(C, ⌧) is ( 1)-representable if for every map X ! G, where
X is a representable object of Shv(C, ⌧), the base change F ⇥ G X is also representable. Let n 0, we say that F 2 Shv(C, ⌧) is n-geometric if it satisfies the following two conditions: (i) It admits an n-atlas, i.e. a morphism p : U ! F from a representable object U such that p is (n 1)representable and it lies in P.

(ii) The diagonal map F ! F ⇥ F is (n 1)-representable.

{Y i ! Y } the pullback F ⇥ Y Y i is representable then so is F .
Remark 5.2.3.6. When the geometric context is closed under ⌧ -descent the definition of a geometric stack becomes simpler since it turns out to be ambiguous to require the representability of the diagonal map.

Example 5.2.3.7. Many examples of geometric contexts can be given but our main object of study will be the geometric context (Afd k , ⌧ ét , P sm ), where ⌧ ét denotes the quasi-étale topology on Afd k , and P sm denotes the collection of quasi-smooth morphisms, see [Ber94a] chapter 3 for the definitions of quasi-étale and quasi-smooth morphisms of k-analytic spaces. Such geometric context is closed under ⌧ ét -descent and we will call the corresponding geometric stacks as k-analytic stacks.

Let G be a smooth group object in the 1-category Shv(C, ⌧). Suppose that G acts on a representable object X. We can form its quotient stack via the (homotopy) colimit of the diagram,

. . . G 2 ⇥ X G ⇥ X X
We denote such (homotopy) colimit by [X/G] and refer it as the stacky quotient of X by G.

Lemma 5.2.3.8. Let (C, ⌧, P) be a geometric context satisfying ⌧ -descent. Let G be a smooth group object in the 1-category Shv(C, ⌧) acting on a representable object X. Then the stacky quotient [X/G] is a geometric stack.

Proof. It suffices to verify condition (1) of Definition 2.12. By definitoin of [X/G] we have a canonical morphism X ! [X/G] which is easily seen to be (-1)-representable and smooth. Therefore, [X/G] is a 0-geometric stack. where the latter consists of the stack of morphisms between BG and BGL an n , considered as ind-pro-stacks. This is not really the case, but it a reasonable conceptual approximation. We will explore this idea in detail using the language of Ind(Pro(S))-enriched 1-categories.

Moduli of k-lisse sheaves on the étale site of a proper normal scheme

Let X be a proper normal scheme over an algebraically closed field K. Let x : SpecK ! X be a fixed geometric point of X. Thanks to [GR, Theorem 2.9, exposé 10] the étale fundamental group ⇡ ét 1 (X) is topologically of finite presentation. As a consequence, the results proved in the previous §hold true for the profinite group G = ⇡ ét 1 (X). In particular,

LocSys `,n (X) := LocSys `,n (⇡ ét 1 (X))
is representable by a k-analytic stack. In this §, we will show that the moduli LocSys `,n (X) parametrizes proétale local systems of rank n on X. This is a consequence of the fact that the étale fundamental group of X parametrizes étale local systems on X with finite coefficients. As we are interested in the local systems valued in k-affinoid algebras, the pro-étale topology is thus more suited for us. This is dealt with in §3.2. In §3.1 we prove some results concerning perfecteness of étale cohomology chains with derived coefficients. These results are known to experts but hard to locate in the literature so we prefer to give a full account of these as they will be important for us in order to show the existence of the cotangent complex of LocSys `,n (X). Proposition 5.3.1.7. Let A 2 CAlg Z/`nZ be a Noetherian simplicial Z/`nZ-algebra. Let N be a local system of A-modules on X ét with values in the 1-category of perfect A-modules, Perf(A). Then the étale cohomology of the local system N , denoted R (X ét , N), is a perfect A-module.

Remark 5.3.1.8. Note that the statement of Theorem 5.3.1.7 concerns the chain level and not the étale cohomology of the complex R (X ét , N), thus it is a stronger statement than just requiring finiteness of the corresponding étale cohomology groups.

Proof of Theorem 5.3.1.7. Let N be a local system on X ét of perfect A-modules, i.e., there exists an étale covering U ! X, such that N |U ' f ⇤ (P ), where f : U ! SpecK denotes the structural map and P 2 Perf(A) is a perfect A-module. Our goal is to show that R (X ét , N) 2 Perf(A).

By Theorem 5.3.1.4 it suffices to show that for each i 2 Z, the cohomology groups

H i (X ét , N) := ⇡ i (R (X ét , N))
is of finite type over ⇡ 0 (A) and moreover

R (X ét , N) 2 Mod A
is of finite Tor-dimension over A. Without loss of generality we can assume that N is a connective perfect Amodule on the étale site X ét , i.e., the discrete ⇡ 0 (A)-étale sheaves on X, ⇡ i (N ) vanish for i < 0.

Thanks to [GL14, Proposition 4.2.10] and its proof we deduce that ⇡ i (N ) is an étale local system of finitely presented (discrete) ⇡ 0 (A)-modules on X ét , for each i 0. For a fixed integer i 0 the homotopy sheaf ⇡ i (N ) is a local system of finitely presented (discrete) ⇡ 0 (A)-modules. It thus follows that there exists an étale covering

V ! X such that ⇡ i (N ) |V ' g ⇤
E, where g : V ! SpecK denotes the structural map and E denotes a suitable ⇡ 0 (A)-module of finite presentation. As X is a normal scheme we can assume without loss of generality that the étale map V ! X is a Galois covering (in particular it is finite étale). It follows by Galois descent that

R (X ét , ⇡ i (N )) ' R (G, R (X ét , g ⇤ E)),
where G is the finite group of automorphisms of the Galois covering V ! X. Assume first that R (X ét , g ⇤ E) is an A-module whose homotopy groups are finitely generated over ⇡ 0 (A). Since the group G is finite, the group cohomology of G with Z-coefficients is finitely generated and of torsion, we thus conclude by the corresponding Grothendieck spectral sequence that the homotopy groups of the complex

R (G, R (X ét , g ⇤ E)) 2 Mod ⇡0(A)
are finitely generated over ⇡ 0 (A). We are thus reduced to the case where ⇡ i (N ) is itself a constant ⇡ 0 (A)module on X ét . By the projection formula we can reduce to the case where ⇡ i (N ) is ⇡ 0 (A) itself. Again by the projection formula we can reduce to the case where ⇡ 0 (A) ' Z/`nZ in which case the result follows readily by Theorem 5.3.1.1. By induction on the Postnikov tower associated to N we conclude that given n 0 we have a fiber sequence of étale A-modules,

⌧ n+1 N ! ⌧ n N ! ⇡ n+1 (N )[n + 2],
such that, by our inductive hypothesis both complexes R (X ét , ⌧ n N ) and R (X ét , ⇡ n+1 (N ))[n + 2] 2 Mod A Proof. Let A 0 be a formal model for A. Note that A 0 is an open subring of A which is p-adically complete and therefore is a pro-discrete ring implying that the group GL n (A 0 ) is a pro-discrete group as in [Noo04, Definition 2.1], thanks to [BS13,Lemma 7.4.6] the result follows if we replace A by A 0 in the statement of the Lemma. Let,

⇢ : ⇡ ét 1 (X) ! GL n (A),
be a continuous representation and

U = ⇢ 1 (GL n (A 0 )), note that U is an open subgroup of ⇡ ét 1 (X), therefore it defines a pointed covering X U ! X with ⇡ ét 1 (X U ) = U . The induced representation, ⇡ ét 1 (X U ) ! GL n (A 0
), defines thus an element M 2 Loc X U (A 0 ) and hence, by inverting p,it produces a local system M 0 2 Loc X U (A). Such element M 0 comes equipped with descent data for X U ! X and therefore comes from a unique N (⇢) in Loc X (A). Conversely, fix some N 2 Loc X (A) which, for suitable n, we can see it as a F GLn(A) -torsor, which is a sheaf for the pro-étale topology on X via [BS13, Lemma 4.2.12], here F GLn(A) denotes the sheaf on X proét defined informally via, T 2 X proét 7 ! Map cont (T, GL n (A)). Let S 2 GL n (A)-Set then we have an induced representation,

⇢ S : F GLn(A) ! F Aut(S) ,
of pro-étale local sheaves. The pushout of N along ⇢ S defines an element N S 2 Loc X with stalk S, which is functorial in S and therefore it defines a functor GL n (A)-Set ! Loc X,n (A) compatible with the fiber functor. By Theorem 5.3.2.3, GL n (A) is Noohi and therefore it is possible to associated it a continuous homomorphism

⇢ N : ⇡ ét 1 (X) ! GL n (A)
, which gives an inverse for the previous construction. This establishes the equivalence of the statement, as desired.

Corollary 5.3.2.5. The non-archimedean stack LocSys `,n (X) represents the functor Afd op k ! S given on objects by the formula,

A 7 ! Loc X,n (A),
where Loc X,n (A) denotes the groupoid of local systems of projective A-modules locally of rank n on the pro-étale topology of X.

Proof. It follows by the construction of quotient stack and Theorem 5.3.2.4.

Moduli of continuous k -adic representations

In this section we prove several results concerning the 1-category of derived continuous k -adic representations and the associated derived moduli stack. Even though such results are somewhat secondary to our main goal they will prove useful in proving the representability of derived moduli stack of rank n continuous k-adic representations.

Preliminaries

Let X 2 Pro(S fc ) be a profinite space which we suppose fixed throughout this §. Assume further that X is connected, i.e.

⇡ 0 Mat(X) ' ⇤
where Mat(X) := Map Pro(S fc ) ⇤, X 2 S.

Definition 5.4.1.1. Let A 2 CAlg ad k and n 1 an integer. We define A n 2 CAlg k n as the derived k n -algebra defined as the pushout of the diagram

A[u] A A A n u7 !p n u7 !0 computed in CAlg k n ,
A ' lim n 1 A n .
Moreover, perfect A-modules are necessarily p-complete and we have an equivalence

M ' lim n 1 (M ⌦ A A n )
in the 1-category Perf(A). Thanks to [Lur16, Proposition 8.1.2.3] it follows that one has an equivalence of 1-categories

Perf(A) ! lim n 1 Perf(A n ).
Therefore, we can (functorially) associate to Perf(A) a pro-object {Perf(A n )} n 2 Pro(Cat 1 ). Construction 5.4.1.3. Let C 2 Cat 1 be an 1-category. The 1-category of pro-objects on C, denoted Pro(C), is defined by means of the following universal property: the 1-category Pro(C) admits small cofiltered colimits and there exists a fully faithful Yoneda embedding j : C ,! Pro(C) such that for any 1-category D admitting small cofiltered colimits we have that pre-composition with j induces an equivalence of 1-categories

Fun 0 Pro(C), D ! Fun C, D
where the left hand side denotes the full subcategory of Fun Pro(C), D spanned by those functors which preserve small cofiltered limits. Moreover, if C is an accessible 1-category which admits finite limits one can give a more explicit of Pro(C) as the full subcategory of Fun(C, S) op spanned by those left exact accessible functors

f : C ! S.
The existence of Pro(C is general (e.g. when C is not necessarily accessible) is guaranteed by [Lur09b, Proposition 5.3.6.2]. We observe that, up to enlarge Grothendieck universes, one can consider the 1-category Pro( \ Cat 1 ) of pro-objects in the 1-category of (not necessarily small) 1-categories, denoted \ Cat 1 .

Remark 5.4.1.4. Since the 1-category S is presentable, we can identify X 2 Pro(S) with a functor

f : S ! S.
Such functor induces a unique, up to contractible indeterminacy, left fibration

F : C ! S obtained as a pullback of the diagram S /f S /⇤ C S f in the 1-category Cat 1 .
Thanks to [Lur09b, Propositino 5.3.2.5], the 1-category S /f is cofiltered. Therefore, the association X 7 ! S /f allow us to interpret X as a pro-system {T i } i , where T i 2 S. Moreover, given X 2 S, we have an equivalence

f (X) ' Map Ti2S /f uT i , X ,
where u : S /f ! S is the forgetful functor.

Fun Pro(Cat1) X, Perf(A) 2 Cat 1 ,
where we consider Perf(A) 2 pro(Cat 1 ). We define also the 1-categories

Mod ad (X)(A) := Ind Perf ad (X)(A) 2 Cat 1 , Coh + (X) ad (A) := Fun Pro(Cat1) X, Coh + (A) 2 Cat 1 Vect(X) ad (A) := Fun Pro(Cat1) X, Vect(A) 2 Cat 1 ,
where Vect(A) ✓ Perf(A) denotes the full subcategory spanned by free A-modules.

Remark 5.4.1.7. The 1-category Perf ad (X)(A) can be identified with

Perf ad (X)(A) ' Fun Pro(Cat1) X, Perf(A) ' lim n 1 Fun Pro(Cat1) X, Perf(A n ) ' lim n 1 colim Xi2S /f X i , Perf(A n )
in the 1-category Cat 1 .

Remark 5.4.1.8. By construction, the 1-category Mod ad (X)(A) is compactly generated and the compact objects span the full subcategory Perf ad (X)(A) ✓ Mod ad (X)(A).

Definition 5.4.1.9. Let C be an additive symmetric monoidal 1-category. Let R 2 CAlg be a commutative derived ring and consider its derived 1-category of modules Mod R 2 Cat ⌦ 1 . We say that C is equipped with an A-linear action if there exists a finite direct sum preserving symmetric monoidal functor

F : Mod ↵ R ! C see [Lur16, Definition D.1.1.
1] for a definition. If C is presentable, then the datum of a linear R-action is equivalent to the existence of a colimit preserving symmetric monoidal functor

F : Mod cn R ! C see [Lur16, Remark D.1.1.5].
If moreover, C is presentable and stable, then the datum of a linear R-action on C is equivalent to give a colimit preserving monoidal functor

F : Mod R ! C.
Proposition 5.4.1.10. Let A 2 CAlg ad k . The 1-category Mod ad (X)(A) is a symmetric monoidal presentable A-linear stable 1-category. The 1-categories Coh + (X) ad (A) and Perf ad (X)(A) are both symmetric monoidal A-linear idempotent complete stable 1-categories and the former admits a canonical t-structure. The 1-category Vect ad (X)(A) is symmetric monoidal, admits an A-linear action and it is moreover additive.

Thus we obtain a coCartesian fibration

D I ! I op ⇥ CAlg ad k ⇥ N op
whose fiber at (i, A, n) 2 I op ⇥ CAlg ad k ⇥ N op can be identified with the 1-category

Fun h(i), Perf(A n ) 2 Cat 1 .
Furthermore, the composition

D I ! CAlg ad k ! N op ,
under the natural projection

I op ⇥ CAlg ad k ! N op ! CAlg ad k ! N op , is a coCartesian fibration whose fiber at (A, n) is naturally equivalent to colim i2I op Fun h(i), Perf(A n ) .
Unstraightening produces a functor

F : CAlg ad k ⇥ N op ! Cat 1
given on objects by the formula

(A, n) 7 ! colim i2I op Fun h(i), Perf(A n ) 2 Cat 1 .
Composing it with the projection functor CAlg ad k ⇥ N op ! N op produces a coCartesian fibration

D I ! N op .
Consider the 1-category of coCartesian sections

Map [ N op , D I 2 Cat 1 .
We have a canonical functor

g : Map [ N op , D I ! Map [ N op , CAlg ad k ⇥ N op ' CAlg ad k .
The coCartesian fibration g produces a well defined functor, up to contractible indeterminacy,

Perf ad (X) : CAlg ad k ! Cat 1
given on objects by the formula

A 2 CAlg ad k 7 ! Perf ad (X)(A) 2 Cat 1 .
Similarly, we can define functors

Mod ad (X), Coh + (X) ad , Vect ad (X) : CAlg ad k ! Cat 1
given on objects by the formulas

A 2 CAlg ad k 7 ! Mod ad (X)(A) 2 Cat 1 , A 2 CAlg ad k 7 ! Coh + (X) ad (A) 2 Cat 1 , A 2 CAlg ad k 7 ! Vect ad (X)(A) 2 Cat 1 ,
respectively.

which we denote simply by ⌦(p m (⇢)), can be identified with

⌦(p M (⇢)) ' ' fib Id⇢ Map Coh + (X) ad (A M ) ⇢ ⌦ (A M ), ⇢ ⌦ (A M ) ! Map Coh +
(X) ad (A) ⇢, ⇢ . We denote the latter object simply by Map /⇢ ⇢ ⌦ A (A M ), ⇢ ⌦ A (A M ) . Since the underlying A-module of ⇢ ⌦ A (A M ) can be identified with ⇢ ⇢ ⌦ A M , we have a chain of natural equivalences of mapping spaces of the form

Map /⇢ ⇢ ⌦ A (A M ), ⇢ ⌦ A (A M ) ' ' Map Coh + (X)(A) /⇢ ⇢ ⌦ A (A M ), ⇢ ⌦ A (A M ) ' Map Coh + (X)(A) /⇢ ⇢, ⇢ ⌦ A (A M )
where the latter mapping space is pointed at the zero morphism. Since ⇢ 2 Perf ad (X)(A) is a dualizable object we have an identification of mapping spaces

Map Coh + (X) ad (A) ⇢, ⇢ ⌦ A M ' Map Coh + (X) ad (A) ⇢ ⌦ ⇢ _ , M .

Consider teh pullback diagram of derived k -adic extensions

A M A A A M [1]
in the 1-category CAlg ad k . As Perf ad (X) is cohesive and the right adjoint ( ) ' : Cat 1 ! S commutes with limits we obtain a pullback diagram of the form

Perf ad (X)(A M ) Perf ad (X)(A) Perf ad (X)(A) Perf(A M [1])
in the 1-category S. By taking fibers at ⇢ 2 Perf ad (X)(A) we have a pullback diagram of spaces

p M (⇢) ⇤ ⇤ p M [1] (⇢)
in the 1-category S. By our previous computations, replacing M with the shift M [1] produces the chain of equivalences

p M (⇢) ' ' ⌦(p M [1] (⇢)) ' Map Coh + (X) ad (A) ⇢ ⌦ ⇢ _ , M[1] ,
as desired.

Definition 5.4.2.8. Let X 2 Pro(S fc ) be a connected profinite space. Let A 2 CAlg ad k and ⇢ 2 Perf ad (X)(A). We say that X is locally p-cohomologically perfect at ⇢ if the object

Map Perf ad (X)(A) 1, ⇢ 2 Sp
where 1 2 Perf ad (X)(A) denotes the unit for the symmetric monoidal strucutre, equipped with its canonical A-linear action is equivalent to a perfect A-module. We say that X is cohomologically perfect if it is locally cohomologically perfect for every ⇢ 2 Perf ad (X)(A) for every admissible derived k -adic algebra A 2 CAlg ad k .

Proposition 5.4.2.9. Let X 2 Pro(S fc ) be a p-cohomologically perfect profinite space. Then for every A 2 CAlg ad k and every ⇢ 2 Perf ad (X)(A) the functor

F : Coh + (A) ! S
given informally via the association

M 2 Coh + (A) 7 ! fib ⇢ Perf ad (X)(A M ) ! Perf ad (X)(A) 2 S is corepresentable by the A-module Map Perf ad (X)(A) 1, ⇢ ⌦ ⇢ _ [1] _ 2 Mod A
Proof. We first prove the following assertion: let C be an A-linear stable presentable 1-category, and C 2 C denote a compact object of C. Then for every object M 2 Mod A we have an equivalence

Map C C, M ' Map C C, 1 C ⌦ A M, in the 1-category Mod A
, where 1 C denotes the unit for symmetric monoidal structure on C. Let D ✓ Mod A denote the full subcategory spanned by those A-modules M such that the assertion holds true. Clearly A 2 D. Since A 2 Mod A generates the 1-category Mod A under small colimits, it suffices to show that D is closed under small colimits. Suppose that

M ' colim i2I M i ,
such that M i 2 D and I is a filtered 1-category. Then by our compactness assumption it follows that we have a chain of equivalences

Map C (C, M ) ' Map C C, colim i M i ' colim i Map C C, M i ' colim i Map C C, 1 ⌦ A M i ' Map C C, 1 ⌦ A (colim i M i ) ' Map C C, 1 C ⌦ A M.
Thus D is closed under filtered colimits. It suffices to show to show then that D is closed under finite colimits. Since Mod A is a stable 1-category it suffices to show that D is closed under finite coproducts and cofibers. Let f : C ! D be a morphism in D, we wish to show that cofib(f )D. Thanks to [Lur12c, Theorem 1.1.2.14] we have an equivalence cofib(f ) ' fib(f ) [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. As a consequence, we can write

Map C C, cofib(f ) ' Map C C, fib(f )[1] ' fib Map C C, f [1] ' cofib Map C C, f ' Map C C, 1 ⌦ A cofib(f ).
The case of coproducts follows along the same lines and it is easier. From this we conclude that D ' Mod A , as desired. In our case, let C := Mod ad (X)(A) and observe that the assertion implies that for every M 2 Mod A we have a chain of equivalences

Map Mod ad (X)(A) ⇢ ⌦ ⇢ _ , M[1] ' Map Mod ad (X)(A) ⌦(⇢ ⌦ ⇢ _ ), 1 ⌦ A M Map Mod ad (X)(A) 1, ⇢ ⌦ ⇢ _ [1] ⌦ A M.
Furthermore, it follows by our hypothesis on X that

C := Map Mod ad (X)(A) 1, ⇢ ⌦ ⇢ _ [1] 2 Mod A
is a perfect A-module. Thus we have a chain of equivalences

Map Mod A C _ , M ' Map Mod A A, C ⌦ A M Map Perf ad (X)(A) 1, (⇢ ⌦ ⇢) _ ⌦ A M Map Perf ad (X)(A) ⇢ ⌦ ⇢ _ , M[1]
and the result now follows by Theorem 5.4.2.9.

Enriched 1-categories

In this §we will state and prove the results in the theory of enriched 1-categories that will prove to be more useful for us in the body of the present text. We will follow mainly the expositions presented in [GH15] and [Lur12c, §4.2].

Preliminaries on Pro(S) and Ind(Pro(S))-enriched 1-categories

We will moreover be more interested in the case where the enrichments are over the 1-categories Pro(S) and Ind(Pro(S)) of pro-objects and ind-pro-objects in S, respectively. The following remark states that both definitions [Lur12c, Definition 4. We start with a general Lemma which will be helpful for us: Lemma 5.5.1.2. Let V ⌦ be a presentably symmetric monoidal 1-category. Suppose we are given a small diagram

F : I ! Cat 1 (V ⌦
). Then the limit C := lim I F exists in the 1-category Cat 1 (V ⌦ ). Furthermore, given any two objects x, y 2 C we have an equivalence of mapping objects

C(x, y) ' lim i2I C i (x i , y i ) 2 V ⌦
where C i := F (i), for each i 2 I and x i , y i denote the images of both x and y under the projection functor C ! C i , respectively.

Proof. We use the notations of [GH15]. In this case, we have a chain of equivalences in

V ⌦ C(x, y) ' ' Map ⇤ `⇤/Cat1(V ⌦ ) E 1 , C) ' lim i2I Map ⇤ `⇤/Cat1(V ⌦ ) E 1 , C i ' lim i2I C i (x i , y i )
where Map denotes the internal mapping object in V ⌦ . This finishes the proof of the statement.

Lemma 5.5.1.6. Let X 2 Pro(S fc ) be a connected profinite space. Given A 2 CAlg ad k and M 2 Perf(A) the fiber of

⇡ A : : Fun Cat1(Pro(S)) X, Perf(A) ! Perf(A) over M 2 Perf(A) is canonical equivalent to the space

Map

Mon E 1 (Pro(S)) ⌦X, End(M ) 2 S.

Proof. Let M 2 Perf(A). The fiber of ⇡ A over A is equivalent to

Map Pro(S) X, BEnd(M ) 2 S
and applying May's Theorem together with the limit colimit formula for mapping spaces in Pro(S) we obtain the canonical equivalence

Map Pro(S) X, BEnd(M ) ' Map Mon E 1 (Pro(S)) ⌦X, End(M )
of mapping spaces. The result now follows.

Construction 5.5.1.7. The fully faithful embedding

S ,! Pro(S)
induces a fully faithful functor

Cat 1 ! Cat 1 Pro(S)
which can be extended, via the universal property of pro-completion, by cofiltered limits to a functor

F : Pro(Cat 1 ) ! Cat 1 Pro(S) .
Moreover, given A 2 CAlg ad k the essential image of Perf(A) 2 Pro(Cat 1 ) by F can be identified with Perf(A) 2 Cat 1 (Pro(S)) as in Theorem 5.5.1.4 as the latter can be identified with

Perf(A) ' lim n 1 Perf(A n ) 2 Cat 1 (Pro(S))
and by construction F preserves cofiltered limits. Similarly, we have a commutative diagram of the form

Pro(S)

Pro(Cat 1 )

Cat 1 Pro(S) F and for this reason X 2 Pro(S) ✓ Pro(Cat 1 ) is sent via F to X 2 Cat 1 (Pro(S)
). As a consequence the functor

F : Pro(Cat 1 ) ! Cat 1 (Pro(S)
) induces a well defined functor

A : Fun Pro(Cat1) X, Perf(A) ! Fun Cat1(Pro(S)) X, Perf(A) .
Moreover, A is functorial in A and thus produces a well defined morphism

: Perf ad (X) ! Fun Cat1(Pro(S)) X, Perf( ) in the 1-category Fun CAlg ad k , Cat 1 .
We have now two potential definitions for the moduli stack Perf ad (X), namely the one provided in the previous §and the second one by defining it via the Pro(S)-enriched 1-categories approach. The following result implies that there is no ambiguity involved in chosen one of these:

Proposition 5.5.1.8. Let X 2 Pro(S fc
) be a connected profinite space. Then the functor

: Perf ad (X) ! Fun Cat1(Pro(S)) X, Perf( )
is an equivalence in the 1-category Fun(CAlg ad k , Cat 1 ). Before giving a proof of Theorem 5.5.1.8 we need a preliminary lemma: Lemma 5.5.1.9. Let X 2 Pro(S) be a connected pro-space. Then X 2 Pro(S 1

), where the latter denotes the 1-category of pro-objects in the 1-category of connected spaces, S 1 . Proof. Let X 2 Pro(cS) be indexed by a cofiltered 1-category I. More explicitly, we can identify X with

X ' lim i2I X i
for suitable X i 2 S. By construction, for each i 2 I, we have an induced morphism i : X ! X i .

By our hypothesis on X" we conclude that each i should factor through a connected component X i ✓ X i . We can thus form the pro-system {X i } i2I 2 Pro(S), which lies in the essential image of the inclusion functor

Pro(S 1

) ,! Pro(S).

Our goal is to show that the induced maps X ! X i induce an equivalence

X ' {X i } i2I
in the 1-category Pro(S). It suffices to show that we have an equivalence of mapping spaces

✓ : lim i2I Map S Y, X i ' lim i2I Map S Y, X i
for every connected space Y 2 S 1 . Notice that ✓ is the cofiltered limit of monomorphisms in the 1-category S.

Thus it is itself a monomorphism. It suffices to show that it is also an effective epimorphism in S.

Let ⇤ ! Y be the unique, up to contractible indeterminacy, morphism in the 1-category S. For each i 2 I, consider also the canonical map lim i2I

Map

S Y, X i ! X i induced by i . Such morphism must necessarily factor through X i ✓ X i , by our choice of X i . It now follows that ⇡ 0 (✓) is surjective. Consequently, the morphism ✓ is an effective epimorphism. The result now follows.

Proof of Theorem 5.5.1.8. Let A 2 CAlg ad k . Both the 1-categories

Perf ad (X)(A), Fun Cat1(Pro(S)) X, Perf(A) 2 Cat 1
are fibered over Perf(A) and we have a commutative diagram of the form

Perf ad (X)(A) Fun Cat1(Pro(S)) X, Perf(A) Perf(A) A (5.5.1.1)
of coCartesian fibrations over Perf(A). Therefore, it suffices to show that for each M 2 Perf(A) the fiber products over M of both vertical functors displayed in (5.5.1.1) are equivalent via A . Thanks to Theorem 5.5.1.9 together with limit-colimit formula for mapping spaces on 1-categories of pro-objects it follows that we can identify the fiber of the left hand side with Map Pro(S) X, BEnd(M ) . The result now follows due to Theorem 5.5.1.6.

ev(⇤) := Perf `(X )(A) ! Perf(A ⌦ k k) at M 2 Perf(A ⌦ k k) is naturally equivalent to the mapping space Map Mon E 1 (Ind(Pro(S))) ⌦X, End(M ) 2 S,
where End(M ) 2 Mon E1 Ind(Pro(S)) denotes the enriched mapping object of M equipped with is multiplicative monoid structure.

Proof. By hypothesis X 2 Pro(S fc ) is connected. Therefore, the fiber of ev(⇤) at M is naturally equivalent to the mapping space

Map

Pro(S⇤) X,

BEnd(M ) ' Map Pro(S⇤) X, BEnd(M 0 ) [p 1 ]. (5.6.1.1)
Consider the usual loop functor ⌦ : S ⇤ ! Mon E1 (S). It induces a canonical functor

⌦ : Pro(S ⇤ ) ! Mon E1 (Pro(S)).
Notice that every transition morphism in the pro-system End(M 0 ) 22 Pro(S) is actually a morphism of monoid objects, i.e. it admits a natural lifting in the 1-category Mon E1 (S). Using the limit-colimit formula for mapping spaces in 1-categories of pro-objects together with the Bar-Cobar equivalence (B, ⌦) we obtain a natural equivalence

Map Pro(S⇤) X, BEnd(M 0 ) [p 1 ] ' Map Mon E 1 (Pro(S)) ⌦X, End(M 0 ) [p 1
]. The universal property of localization at p induces a canonical map

✓ : Map Mon E 1 (Pro(S)) ⌦X, End(M 0 ) [p 1 ] ! Map Mon E 1 (Ind(Pro(S)
)) ⌦X, End(M ) , in the 1-category of spaces S. The result follows if we are able to prove that ✓ is an equivalence. We notice that we cannot apply May delooping theorem component-wise as multiplication by p : End(M 0 ) ! End(M 0 ) is not a morphism of monoid-objects. However, the map ✓ is induced by a morphism of the form

✓ 0 : Map Mon E 1 (Pro(S)) ⌦X, End(M 0 ) ! Map Mon E 1 (Ind(Pro(S)
)) ⌦X, End(M ) . Furthermore, the fiber of the morphism End(M 0 ) ! End(M ) in the 1-category Mon E1 (Ind(Sp pro (p) p 1 ) coincides with the colimit

colim n End(M 0 )/p n [ 1], (5.6.1.2)
which is of p-torsion. Therefore, passing to the filtered colimit along multiplication by p the term in (5.6.1.2) becomes the zero object in the stable 1-category Sp pro (p) p 1 . As a consequence, it follows that ✓ ' ✓ 0 [p 1 ] has contractible fiber and therefore it is an equivalence. Remark 5.6.1.5. Theorem 5.6.1.4 implies that the functor Perf p (X)(A) ! Perf(A ⌦ k k is a coCartesian fibration which corresponds to a functor

F : Perf(A) ! Cat 1
given on objects by the formula

M 2 Perf(A ⌦ k k) 7 ! Map
Mon E 1 (Ind(Pro(S))) ⌦X, End(M ) 2 S. Therefore, we can regard the 1-category Perf p (X)(A) as the 1-category of continuous representations of X with values in perfect A ⌦ k k-modules. Definition 5.6.1.6. We define the Cat 1 -valued functor of p-adic perfect modules on X as the functor

Perf p (X) : CAlg ad k <1 ! Cat 1 ,
given on objects by the formula

A 2 CAlg ad k <1 7 ! Perf p (X)(A) 2 Cat 1 .
An important consequence of Theorem 5.5.2.26 is the following result: which is given on objecs by the formula

Z 2 dAfd <1 k 7 ! Fun ECat1 X, Perf(Z) 2 Cat 1 .
Proof. The result is a direct consequence of the equivalent statement for Perf which is the content of Theorem 5.5.2.26.

Lifting results for continuous p-adic representations of profinite spaces

The following definition is crucial for our purposes:

Definition 5.6.2.1. Let X 2 Pro(S fc
) be a connected profinite space. We say that X is p-cohomologically compact if for any p-torsion Z p -module N 2 Mod Z p with

N ' colim ↵ N ↵ ,
where N ↵ 2 Mod Z for each ↵, we have an equivalence of mapping spaces

Map Mon E 1 (Pro(S)) ⌦X, N ' colim ↵ Map Mon E 1 (Pro(S)) ⌦X, N ↵ ,
i.e. taking continuous cohomology of X iwth torsion coefficients commutes with filtered colimits.

Remark 5.6.2.2. The above definition makes sense when we consider X 2 S. In this case, it is equivalent to ask for a cellular decomposition of X with finitely man cells in each dimension. However X itself might have infinitely many non-zero (finite) homotopy groups.

Example 5.6.2.3.

(i) Suppose Y ! X is a finite morphism in Pro(S fc ), i.e. its fiber is a finite constructible space Z 2 Pro(S fc

). If we assume further that X is p-cohomologically compact, then so it is Y . More generically, the notion of cohomologically compactness is stable under fiber sequences.

(ii) Suppose X 2 Pro(S fc

) is the étale homotopy type of a smooth variety over an algebraically closed field. Then X is cohomologically almost of finite type, see Theorem 5.6.2.4. Let X 2 Pro(S fc ) be a connected profinite space which we assume further to be p-cohomologically compact. Let A 2 (CAlg ad k ) <1 and suppose we are given ⇢ 2 Perf p (X)(A) such that M := ev(⇤)(⇢) 2 Perf(A ⌦ k k) admits a perfect formal model M 0 2 Perf(A). Then there exists Y 2 Mon E1 (Pro(S fc

)) together with a morphism

f : Y ! ⌦X in the 1-category Mon E1 (Pro(S fc
)) such that fib(f ) is finite constructible and such that we have a factorization of the form

Y End(M 0 ) ⌦X End(M ) ⇢ 0 ⇢ in the 1-category Mon E1 (Ind(Pro(S))).
Proof. As X 2 Pro(S fc ) is assumed to be connected, an object ⇢ 2 Perf p (X)(A) corresponds to a morphism

⇢ : ⌦X ! End(M )
in the 1-category Mon E1 (Ind(Pro(S))), where M denotes ev(⇤)(⇢) 2 Perf(A ⌦ k k). We first remark assumption on the existence of a perfect formal model for M , M 0 2 Perf(A) can be dropped, since by [TT90] the trivial square zero extension M ⌦ M [ 1] 2 Perf(A ⌦ k k) admits a formal model and M is a retract of the latter. Therefore, we can replace M with M M [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] and assume from the start that M admits a formal model

M 0 2 Perf(A).
Let n 0 be an integer and consider the truncation functor

⌧ n : Ind(Pro(S)) ! Ind(Pro(S))
induced by the usual truncation functor ⌧ n : S ! S by means of the universal property of both ind and pro completions. More explicitly,

⌧ n (colim i lim j X i, j) ' colim i lim j ⌧ n (X i,j ) 2 Ind(Pro(S)).
As A 2 CAlg ad k is assumed to be truncated and M 0 2 Perf(A) is perfect over A it follows that also M 0 is a truncated A-module. The same conclusion holds for the couple (A⌦ k k, M ). Therefore, there exists a sufficiently large m > 0 such that

⌧ m End(M ) ' End(M ).
(5.6.2.1)

We now proceed to construct such a profinite Y 2 Mon E1 (Pro(S fc )) satisfying the conditions of the statement. Our construction is by means of an inductive reasoning via the relative Postnikov tower of the canonical morphism

g : End(M 0 ) ! End(M )
in the 1-category Mon E1 (Ind(Pro(S))). We start by observing that the cofiber of g is equivalent to the filtered colimit

cofib(g) ' colim k End(M 0 )/p k 2 Ind(Pro(S)).
For this reason, we can identify the fiber of the morphism g with

fib(g) ' colim k End(M 0 )/p k [ 1].
Suppose then n = 0. Consider the pullback diagram

End(M 0 ) 0 ⇡ 0 (End(M 0 )) End(M ) ⇡ 0 (End(M )) q0
in the 1-category Mon E1 (Ind(Pro(S))). By construction, we can identify the fiber of the morphism q : End(M 0

) 0 ! End(M ) with fib(q) ' colim k ⇡ 0 (End(M 0 )/p k )[ 1]).
By the universal of the 0-truncation functor in (5.6.2.1) the composite

⌦X ⇢ ! End(M ) ! ⇡ 0 (End(M ))
in the 1-category Mon E1 (Ind(Pro(S))) factors through a continuous group homomorphism

⇢ 0 : ⇡ 1 (X) ! ⇡ 0 (End(M )),
where the topology on the left hand side group is the profinite one induced from X and the topology on the right hand side group is the topology induced by the ind-pro structure on End(M ). Since ⇡ 0 (End(M 0 )) is an open subgroup of ⇡ 0 (End(M )) and ⇡ 1 (X) is profinite, it follows that the inverse image

⇢ 1 0 (⇡ 0 (End(M 0 )))  ⇡ 1 (X) is of finite index in ⇡ 1 (X). Let U C X be an open normal subgroup such that ⇢ 0 (U ) ✓ ⇡ 0 (End(M 0 )) ✓ ⇡ 0 (End(M ))
and such that ⇡ 1 (X)/U ⇠ = G, where G is a finite group. Consider the pullback diagram

Y 0 U ⌦X ⇡ 1 (X) h0
in the 1-category Mon E1 (Pro(S fc )). By construction, the morphism Y 0 ! ⌦X admits a finite constructible fiber, namely G. Furthermore, we have an equivalence

X ' BY 0 /G
in the 1-category Mon E1 (Ind(Pro(S))). The base step of our inductive reasoning is thus finished. Suppose now that for a given integer n 0 we have constructed a commutative diagram

Y n End(M 0 ) n ⌦X End(M ) hn ⇢
in the 1-category Mon E1 (Ind(Pro(S))), where h n : Y n ! ⌦X admits a finite constructible fiber. Let

g n+1 : End(M 0 ) n+1 ! End(M )
denote the relative (n + 1)-st truncation of the canonical morphism g : End(M 0 ) ! End(M ). We have thus a commutative diagram of the form

End(M 0 ) n+1 End(M 0 ) n . . . End(M 0 ) 0 End(M ) jn jn 1 j0 g0 in the 1-category Mon E1 (Ind(Pro(S))) such that fib(j n ) ' colim k ⇡ n+1 (End(M 0 )/p k )[n + 2]. Consider the following pullback diagram e Y n+1 End(M 0 ) n+1 Y n End(M 0 ) n ⇡n in the 1-category Mon E1 (Ind(Pro(S))). The fiber of the morphism ⇡ n : e Y n+1 ! Y n is equivalent fib(⇡ n ) ' colim k ⇡ n+2 (End(M 0 )/p k )[n + 2].
The fiber sequence

fib(⇡ n ) ! e Y n+1 ! Y n is classified by a morphism ' n : Y n ! colim k ⇡ n+2 (End(M 0 )/p k )[n + 3]
in the 1-category Mon E1 (Ind(Pro(S))). Notice that, ⇡ n + 2(End(M 0 )/p k ) is a discrete group and the monoid structure on ⇡ n+2 (End(M 0 )/p k )[n + 3] is abelian, as n + 3 2. Therefore, the transition maps in the ind-filtered colimit

colim k ⇡ n+2 (End(M 0 )/p k )[n + 3]
do preserve the monoid structure on each term. Thus we find that ' n is actually a morphism in the 1-category Mon E1 (Ind(Pro(S))). As Y n 2 Pro(S fc ) ,! Pro(S) it follows that ' n factors through End(M 0 )/p k for sufficiently large k 1. This induces a fiber sequence of the form

⇡ n+2 (End(M 0 )/p k )[n + 2] ! Y n+1 ! Y n ,
such that we have an induced map

Y n+1 ! End(M 0
) n+1 in the 1-category Mon E1 (Ind(Pro(S))) which coincides with the composite

Y n+1 ! e Y n+1 ! End(M 0 ) n+1 .
As X is p-cohomologically compact and the morphism BY n ! X admits a finite constructible fiber, by assumption, it follows that BY n is also p-cohomologically compact. As ⇡ n+2 (End(M 0 )/p k )[n + 2] is p-torsion over Z p it follows, by Lazard's theorem, that we have an equivalence

⇡ n+2 (End(M 0 )/p k )[n + 2] colim ↵ N ↵ [n + 2],
in the derived 1-category Mod Z p k , where each N ↵ is a finite discrete Z/p k Z-module. We obtain thus, by pcohomological compactness, an equivalence of mapping spaces

Map Mon E 1 (Pro(S fc )) Y n , ⇡ n+2 (End(M 0 )/p k [n + 3] ' ' colim ↵ Map Mon E 1 (Pro(S fc )) Y n , N ↵ [n + 3] .
Therefore, the map ' n above factors through a morphism

' ,n : Y n ! N [n + 3]
in the 1-category Mon E1 (S fc ) and such factorization produces an extension

N [n + 2] ! Y n+1 jn+1 ! Y n ,
in the 1-category Mon E1 (Pro(S fc )). Moreover, by construction, it follows that the composite

Y n+1 ! Y n ! • • • ! ⌦X ! End(M )
factors through the canonical morphism End(M 0 ) n+1 ! End(M ). The inductive step is thus completed. In order to finish the proof of the statement it suffices now to observe that there exists a sufficiently large integer n 1 such that both End(M 0 ), End(M ) are n-truncated objects in the 1-category Ind(Pro(S)). Thus for such n 1 we have an equivalence

End(M 0 ) n+1 ' End(M 0 ).
We have thus produced a finite morphism Y ! ⌦ with Y := Y n+1 and a commutative diagram

Y End(M 0 ) ⌦X End(M )
in the 1-category Mon E1 (Ind(Pro(S))) and the claim is proved.

Remark 5.6.2.5. In the above proof there exists no need to assume M 2 Perf(A ⌦ k k) admits a perfect module M 0 2 Perf(A). Actually, it suffices to assume that there exists a formal model M 0 2 Coh + (A) for M which is truncated. This assumption is always verified. Indeed, there exists a formal model

M 0 2 Coh + (A) for M 2 Perf(A ⌦ k k) thanks to [Ant18b, Proposition A.2.1]. Even if M 0 is not truncated, its rigidification M ⌦ k k is,
as A and thus A ⌦ k k are assumed from the start to be truncated. If we pick m > 0 sufficiently large such that

M is m-truncated then ⌧ m M 0 2 Coh + (A) is still a formal model for M 2 Perf(A ⌦ k k).
Construction 5.6.2.6. Let A 2 CAlg ad k be a truncated derived k -adic algebra. The functor

L p : Sp pro (p) ! Sp pro (p) p 1 induces a base change functor f ⇤ A : Perf ad (X)(A) ! Perf p (X)(A) of stable 1-categories.
Definition 5.6.2.7. We say that ⇢ 2 Perf `(X )(A) is liftable if ⇢ lies in the essential image of the base change functor f ⇤ A as above.

As a consequence of Theorem 5.6.2.4 we have the following Corollary:

Corollary 5.6.2.8. Let X 2 Pro(S fc ) be a connected and p-cohomologically compact profinite space. Consider a truncated derived k -adic algebra A 2 CAlg ad k . Then every object ⇢ 2 Perf p (X)(A) is a retract of a liftable ⇢ 0 2 Perf p (X)(A). Moreover, Perf p (X)(A) is canonically equivalent to the idempotent completion of the base change Perf ad (X)(A) ⌦ k k in the 1-category of stable and idempotent complete small 1-categories, Cat st,idem

1 . Proof. Let ⇢ 2 Perf p (X)(A), M := ev(⇤)(⇢) 2 Perf(A ⌦ k k) and let ⇡ : Y ! X be a finite morphism in Pro(S fc
) with Y connected such that the composite representation

⌦Y ! ⌦X ! End(M )
factors through a morphism End(M 0 ) ! End(M ) where we can suppose that M 0 2 Perf(A). The proof of Theorem 5.6.2.4 implies that we have a commutative diagram of the form

Y ' Y n+1 Y n . . . Y 0 X gn gn 1 j0 g0
in the 1-category Pro(S fc ). Furthermore, we have by construction a canonical equivalence X ' Y 0 / , where is a suitable finite group (not necessarily abelian). In particular, we have an equivalence of 1-categories

Perf p (X)(A) ' Perf p (Y 0 )(A)
of A ⌦ k k-linear stable 1-categories. Moreover, the proof of Theorem 5.6.2.4 implies that for each integer 0  i  n 1 we can choose the morphism

g i : Y i+1 ! Y i such that it is a M i [n + 2]
-torsor for a given finite abelian group M i . As A ⌦ k k lives over a field of characteristic zero, namely k it follows that we have an equivalence of 1-categories

Perf p (Y 0 )(A) ' Perf p (Y i )(A)
for each integer 0  i  n. As a consequence, we deduce that one has an equivalence of 1-categories

Perf p (X)(A) ' Perf p (Y )(A) .

Thus we have an adjunction of the form

⇡ ⇤ : Perf p (X)(A) Perf p (Y )(A) : ⇡ ⇤
where ⇡ ⇤ denotes the restriction functor along ⇡ : Y ! X. Given ⇢ 2 Perf p (Y )(A) we have an equivalence

⇡ ⇤ ⇢ ' ⇢ ⌦ k k[ ],
where k[ ] denotes the free k-algebra on the finite group . The representation ⇢ is a retract of ⇡ ⇤ ⇡ ⇤ ⇢, given by the trivial morphism of groups {1} ! .

Observe further that the representation ⇡ ⇤ ⇡ ⇤ ⇢ is liftable by the choice of Y , as ⇡ ⇤ ⇢ is so. We are thus reduced to prove the second part of the statementn, namely that Perf p (X)(A) is equivalent to the idempotent completion of the A ⌦ k k-linear stable 1-category Perf ad (X)(A) ⌦ k k. It suffices, in fact, to prove that Perf p (X)(A) is idempotent complete. Thanks to Theorem 5.6.2.4 we can assume from the start that ⇢ is liftable from the start. In this case, it suffices to show that for every idempotent

f : ⇢ ! ⇢,
in the 1-category Perf p (X)(A) admits a fiber and cofiber in the f . As in Theorem 5.6.2.5 we can suppose that M 0 has non-trivial homotopy groups concentrated in a finite number of degrees. by Theorem 5.6.2.5 the proof of Theorem 5.6.2.4 applies in this case. In this case, it suffices to show that f admits fiber and cofiber sequences in the 1-category Coh + (X) ad (A), which follows as the latter 1-category is idempotent complete.

1-category Perf p (X)(A). Let M := ev(⇤)(⇢) 2 Perf(A ⌦ k k).
Corollary 5.6.2.9. Let A 2 CAlg ad k be a truncated derived k -adic algebra. Then the 1-category Perf p (X)(A) is stable and admits a natural symmetric monoidal structure.

Proof. Thanks to Theorem 5.6.2.8 and the formula for mapping spaces it follows that Perf p (X)(A) is an A⌦ k klinear 1-category which is equivalent to the tensor product

Perf ad (X)(A) ⌦ k k in the 1-category Cat st,idem 1
. This implies that Perf p (X)(A) is stable and the symmetric monoidal structure on Perf p (X)(A) is induced from the one on Perf ad (X)(A).

Definition 5.6.2.10. Let X 2 Pro(S fc ) and A 2 (CAlg ad k ) <1 . We define Mod p (X)(A) := Ind(Perf p (X)(A)).

Corollary 5.6.2.11. The 1-category Mod p (X)(A) is a presentable stable 1-category which is moreover locally rigid and we have an equivalence of presentable 1-categories

Mod ad (X)(A) ⌦ k k ' Mod p (X)(A) 2 Pr st L .
Proof. Presentability of Mod p (X)(A) follows from Theorem 5.6.2.8. Moreover, we have a chain of equivalences

Mod p (X)(A) ' ' Mod ad (X)(A) ⌦ k k ' Mod A⌦ k k Mod ad (X)(A)
where the latter equivalence follows from [Lur12c, Proposition 6.3.4.6]. As Mod ad (X)(A) is locally rigid it follows also that Mod p (X)(A) is locally rigid, thanks to [Lur16, Lemma D.7.7.2]. The result now follows.

Moduli of derived continuous p-adic representations

In this §we define the moduli of derived continuous p-adic representations of a profinite space X and we show that it admits a derived structure under certain mild assumptions on the profinite space X 2 Pro(S fc

). Definition 5.6.3.1. Let X 2 Pro(S fc

). The moduli of derived continuous p-adic representations of X is defined as the right Kan extension along the canonical inclusion functor

j : dAfd <1 k ! dAfd k of the moduli functor PerfSys `(X ) := ( ) ' Perf p (X) : dAfd <1 k op ! S
which is given on objects by the formula

Z 2 dAfd <1 k 7 ! Perf p (X)( (Z)) ' 2 S.
The following result is a reality check:

Lemma 5.6.3.2. Let Z 2 dAfd k . Then we have a natural equivalence

PerfSys `(X )(Z) ' lim n PerfSys `(X )(t n Z)
in the 1-category S. In particular, the functor PerfSys `(X ) is nilcomplete.

Proof. This statement was stated without proof in [GR14, p. 10]. Let T X := (dAfd <1 k ) op Z/ and denote T 0 Z the full subcategory of T Z spanned by those objects of the form t n ! Z, for each n 0. By the end formula for right Kan extensions it suffices to show that the inclusion functor T 0 Z ! T Z is a final functor. Thanks to the dual statement of [Lur09b, Theorem 4.1.3.1] it suffices to show that for every

(Y ! Z) op in T Z , the 1-category (T 0 Z ) /Y
has weakly contractible enveloping groupoid. We can identify the 1-category (T 0 Z ) /Y with the 1-category of factorizations of the morphism (Y ! Z) op . Thanks to the universal property of n-th truncations and the fact that Y is a truncated derived k-affinoid space it follows that there exists a sufficiently large integer m such that (Y ! X) op factors uniquely (up to contractible indeterminacy) as (Y ! t m X ! X) op .

Therefore the 1-category (T 0 X ) /Y is cofiltered and thus weakly contractible, as desired. Proposition 5.6.3.3. The functor PerfSys `(X ) : dAfd op k ! S satisfies étale hyper-descent. Proof.

Proposition 5.6.3.4. The functor PerfSys `(X ) : dAfd op k ! S is cohesive. Proof. The right adjoint ( ) ' : Cat 1 ! S commutes with small limits and in particular with finite limits. Moreover, PerfSys `(X ) is nilcomplete, thus we can restrict ourselves to prove the assertion when restricted to truncated objects. As a consequence, it suffices to show that the functor

Perf p (X) : dAfd <1 k ! Cat 1 is infinitesimally cartesian. Let Z 2 dAfd <1 k and let d : L an Z ! M be a k-analytic derivation of Z, with M 2 Coh + (A).
Thanks to [Ant18b, Theorem A.2.1] we can lift d to a formal derivation

d 0 : L ad A ! M 0 in the 1-category Coh + (A)
where A 2 CAlg ad k is a formal model for Z 2 dAfd k which we can assume to be truncated. In this case, the canonical functor

Perf p (X)(A d 0 [M 0 ]) ! Perf p (X)(A) ⇥ Perfp(X)(A⌦M 0 ) Perf p (X)(A)
is an equivalence, which follows immediately from Theorem 5.5.2.16.

We shall review the main definitions:

Definition 5.7.1.4. Let F 2 dSt dAfd k , ⌧ ét . We say that F admits a global analytic cotangent complex if the following two conditions are verified:

(i) Given Z 2 dAfd k and z : Z ! F a morphism, the functor

Der an F (Z, ) : Coh + (Z) ! S
given on objects by the formula

M 7 ! fib z F (Z[M ]) ! F (Z) ,
is corepresented by an eventually connective object L an F,z 2 Coh + (Z).

(ii) For any morphism f : Z ! Z 0 in the 1-category dAfd k and any morphism z 0 : Z ! F we have a canonical equivalence,

f ⇤ L an F,z 0 ' L an F,z
where z := z 0 f . Definition 5.7.1.5. Let F 2 dSt(dAfd k , ⌧ ét ). We say that F is cohesive if for every Z 2 dAfd k and every coherent sheaf F 2 Coh 1 (Z) together with a derivation

d : L an X ! F the natural map F Z d [F[ 1]] ! F (Z) ⇥ F (Z[F]) F (Z) is an equivalence in the 1-category S.
Definition 5.7.1.6. We say that F 2 dSt dAfd k , ⌧ ét . We say that F is convergent if for every derived k-affinoid space Z the canonical morphism,

F (Z) ! lim n 0 F (t n Z),
is an equivalence in the 1-category S.

Main results

Let X be a proper and smooth scheme over an algebraically closed field. To such X we can associate it a profinite space, namely its étale homotopy type one might ask how ⇢ can be deformed into a continuous representation ⇢ : ⇡ ét 1 (X) ! GL n (Q `). This amounts to understand the formal moduli problem Def ⇢ : CAlg sm F `! S given on objects by the formula

A 2 CAlg sm 7 ! Map cont (Sh ét (X), BGL n (A)) ⇥ Map cont (Shét(X),BGLn(F `)) {⇢} 2 S,
where Sh ét (X) 2 Pro S fc denotes the étale homotopy type of X. Given ⇢ as above, the functor Def ⇢ was first considered by Mazur in [START_REF] Mazur | Deforming galois representations[END_REF], for Galois representations, in the discrete case. More recently, Galatius and Venkatesh studied its derived structure in detail, see [START_REF] Galatius | Derived Galois deformation rings[END_REF] for more details. One can prove, using similar methods to those in [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] that the tangent complex of Def ⇢ is naturally equivalent to T Def⇢ ' C ⇤ ét X, Ad(⇢) [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF],

in the derived 1-category Mod F `. We can consider Def ⇢ as a derived W (F `)-adic scheme which is locally admissible, in the sense of [START_REF] António | p-adic derived formal geometry and derived Raynaud localization Theorem[END_REF]. Therefore, one can consider its rigidification

Def rig ⇢ 2 dAn Q `.
By construction, we have a canonical inclusion functor

j ⇢ : Def rig ⇢ ! LocSys `,n (X).
By comparing both analytic cotangent complexes, one arrives at the following result: Proposition 6.1.1.2. The morphism of derived stacks

j ⇢ : Def rig ⇢ ! LocSys `,n (X)
exhibits Def rig ⇢ as an admissible open substack of LocSys `,n (X).

Theorem 6.1.1.2 implies, in particular, that LocSys `,n (X) admits as an admissible analytic substack the disjoing union `⇢ Def rig ⇢ , indexed by the set of contininuous representations ⇢ : ⇡ ét 1 (X) ! GL n (Q `). Nonetheless, the moduli LocSys `,n (X) admits more (analytic) points in general than those contained in the disjoint union `⇢ Def rig ⇢ . This situation renders difficult the study of trace formulas on LocSys `,n (X) which was the first motivation for the study of such moduli. Ideally, one would like to "glue" the connected components of LocSys `,n (X) in order to have a better behaved global geometry. More specifically, one would like to exhibit a moduli algebras or analytic stack M `,n (X) of finite type over Q `such that the space closed points M `,n (X)(Q `) 2 S would correspond to continuous `-adic representations of ⇡ ét 1 (X). Moreover, one should expect such moduli stack to have a natural derived structure which would provided an understanding of deformations of `-continuous representations ⇢.

Such principle has been largely successful for instance in the context of continuous p-adic representations of a Galois group of a local field of mixed characteristic (0, p). Via p-adic Hodge structure and a scheme-image construction provided in [START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF], the authors consider the moduli of Kisin modules which they prove to be an indalgebraic stack admitting strata given by algebraic stacks of Kisin modules of a fixed height. Unfortunately, the methods used in [START_REF] Emerton | Scheme-theoretic images" of morphisms of stacks[END_REF], namely the scheme-image construction, do not directly generalize to the derived setting. Recent unpublished work of M. Porta and V. Melani regarding formal loop stacks might provide an effective answer to this problem, which we pretend to explore in the near future. However, to the best of the author's knowledge, there is no other successful attempts outside the scope of p-adic Hodge theory.

We will also study the existence of a 2 2d-shifted symplectic form on LocSys `,n (X), where d = dim X. Even though LocSys `,n (X) is not an instance of an analytic mapping stack it behaves as such. We need to introduce the moduli stack PerfSys `(X ) which corresponds to the moduli of objects associated to the Cat st,!,⌦ 1 valued moduli stack given on objects by the formula

Z 2 dAfd Q `7 ! Fun ECat1 |X| ét , Perf (Z)
where ECat 1 denotes the 1-category of (small) Ind(Pro(S))-enriched 1-categories. We are then able to prove: 6.2 Setting the stage

Recall on the monodromy of (local) inertia

In this subsection we recall some well known facts on the monodromy of the local inertia, our exposition follows closely [9, §1.3].

Let K be a local field, O K its ring of integers and k the residue field which we assume to be of characteristic p > 0 different from `. Fix K an algebraic closure of K and denote by G K := Gal K/K its absolute Galois group.

Definition 6.2.1.1. Given a finite Galois extension L/K with Galois group Gal (L/K) we define its inertia group, denoted I L/K , as the subgroup of Gal (L/K) spanned by those elements of Gal (L/K) which act trivially on l := O L /m L , where L denotes the ring of integers of L and m L the corresponding maximal ideal.

Remark 6.2.1.2. We can identify the inertia subgroup I L/K of Gal(L/K) with the kernel of the surjective continuous group homomorphism q : Gal(L/K) ! Gal(l/k). We have thus a short exact sequence of profinite groups

1 ! I L/K ! Gal(L/K) ! Gal(l/k) ! 1. (6.2.1.1)
In particular, we deduce that the inertia subgroup I L/K can be identified with a normal subgroup of Gal(L/K).

Remark 6.2.1.3. Letting the field extension L/K vary, we can assemble together the short exact sequences displayed in (6.2.1.1) thus obtaining a short exact sequence of profinite groups

1 ! I K ! G K ! G k ! 1, (6.2.1.2)
where G k := Gal(k/k) where k denotes the algebraic closure of k determined by K. Definition 6.2.1.4 (Absolute inertia). Define the (absolute) inertia group of K as the inverse limit

I K := lim L/K finite I L/K ,
which we canonically identify with a subgroup of G K . Definition 6.2.1.5 (Wild inertia). Let L/K be a field extension as above. We let P L/K denote the subgroup of I L/K which consists of those elements of I L/K acting trivially on O L /m 2 L . We refer to P L/K as the wild inertia group associated to L/K. Definition 6.2.1.6 (Absolute wild inertia). We define the absolute wild inertia group of K as: P K := lim L finite P L/K . Remark 6.2.1.7. We can identify the absolute wild inertia group P K with a normal subgroup of I K .

Consider the exact sequence

1 ! P K ! I K ! I K /P K ! 1. (6.2.1.3)
Thanks to [START_REF] De Jong | Stacks Project[END_REF]Lemma 53.13.6] it follows that the wild inertia group P K is a pro-p group. When K = Q p a theorem of Iwasawa implies that P K is not topologically of finite generation, even though G K is so. Nonetheless, the quotient I K /P K is much more amenable: Proposition 6.2.1.8. [4, Corollary 13] Let p := char(k) denote the residual characteristic of K. The quotient

I K /P K is canonically isomorphic to b Z 0 (1)
, where the latter denotes the profinite group

Q q6 =p Z q (1)
. In particular, the quotient profinite group I K /P K is topologically of finite generation.

Define P K,`t o be the inverse image of Q q6 =`,p Z q in I K . We have then a short exact sequence of profinite groups

1 ! P K ! P K,`! Y q6 =`,p Z q ! 1.
Define similarly G K,`: = G K /P K,`t he quotient of G K by P K,`. We have a short exact sequence of profinite groups

1 ! P K,`! G K ! G K,`! 1.
(6.2.1.4)

Assembling together (6.2.1.3) and Theorem 6.2.1.8 we obtain a short exact sequence

1 ! Z `(1) ! G K,`! G k ! 1.
(6.2.1.5) Remark 6.2.1.9. As a consequence of both (6.2.1.4) and (6.2.1.5) the quotient G K,`i s topologically of finite type.

Suppose we are now given a continuous representation

⇢ : G K ! GL n (E `),
where E `denotes a finite field extension of Q `. Up to conjugation, we might assume that ⇢ preserves a lattice of E `. More explicitly, up to conjugation we have a commutative diagram of the form

G K GL n (Z `) GL n (Q `) e ⇢ ⇢ .
Therefore e ⇢ (G K ) is a closed subgroup of GL n (Z `). Consider the short exact sequence

1 ! N 1 ! GL n (Z `) ! GL n (F `) ! 1,
where N 1 denotes the group of GL n (Z p ) formed by congruent to Id mod `matrices. In particular, N 1 is a profinite pro-`group. By construction, every finite quotient of P K,`i s of order prime to `. One then has necessarily

⇢ (P K,`) \ N 1 = {1}.
As a consequence, the group ⇢(P K,`) injects into the finite group GL n (F `) under ⇢. Which in turn implies that the (absolute) wild inertia group P K itself acts on GL n (Q `) via a finite quotient.

Geometric étale fundamental groups

Let X be a geometrically connected smooth scheme over an algebraically closed field k of positive characteristic. Fix once and for all a geometric point ◆ x : x ! X and consider the corresponding étale fundamental group ⇡ ét 1 (X) := ⇡ ét 1 (X, x), a profinite group. If we assume that X is moreover proper one has the following classical result: Theorem 6.2.2.1. [12, Exposé 10, Thm 2.9] Let X be a smooth and proper scheme over an algebraically closed field. Then its étale fundamental group ⇡ ét 1 X is topologically of finite type. Unfortunately, the statement of Theorem 6.2.2.1 does not hold in the non-proper case as the following proposition illustrates: Proposition 6.2.2.2. Let k be an algebraically closed field of positive characteristic. Then the étale fundamental group of the affine line ⇡ ét 1 (A 1 k ) is not topologically finitely generated. Proof. For each integer n 1, one can exhibit Galois covers of A 1 k whose corresponding automorphism group is isomorphic to (Z/pZ) n . This statement readily implies that ⇡ ét 1 (A 1 k ) does not admit a finite number of topological generators. In order to construct such coverings, we consider the following endomorphism of the affine line

n : A 1 k ! A 1 k , defined via the formula n : x 7 ! x p n
x. The endormophism n respects the additive group structure on A 1 k . Moreover, the differential of n equals 1. For this reason, n induces an isomorphism on cotangent spaces and, in particular, it is an étale morphism. As k is algebraically closed, n is surjective and it is finite, thus a finite étale covering. The automorphism group of n is naturally identified with its kernel, which is isomorphic to F p n . The statement of the proposition now follows. (X) the étale shape of X defined as the fundamental groupoid associated to the 1-topos Shv ét (X) ^, of hyper-complete étale sheaves on X. Definition 6.3.1.4. Let X be as above. We define the derived moduli stack of `-adic pro-étale local systems of rank n on X as the functor RLocSys `,n (X) : dAfd op Q `! S, given informally on objects by the formula

Z 2 dAfd op Q `7 ! lim n 0 Map Mon E 1 (C) ⇣ Sh ét (X), BEnd (t n (Z)) ⌘ ,
where t n (Z) denotes the n-th truncation functor on derived Q `-affinoid spaces.

Notation 6.3.1.5. Given Z 2 dAfd Q `we sometimes prefer to employ the notation RLocSys `,n (X)( (Z)) := RLocSys `,n (X)(Z).

Let ⇢ 2 RLocSys `,n (X)( (Z)), we refer to it as a continuous representation of Sh ét (X) with coefficients in (Z). Given Z 2 Afd op , the object BEnd(Z) 2 Ind Pro S is 1-truncated. As a consequence, we have an equivalence of mapping spaces:

Map

Ind(Pro(S)) Sh ét (X), BEnd(Z) ' Map

Ind(Pro(S)) ⇣ ⌧ 1 Sh ét (X), BEnd(Z) ⌘ .
We have moreover an equivalence of profinite spaces ⌧ 1 Sh ét (X) ' B⇡ ét 1 (X). Given a continuous group homomorphism ⇢ : ⇡ ét 1 (X) ! GL n (A) we can associate, via the cobar construction performed in the 1-category Top na , a well defined morphism B⇢ : B⇡ ét 1 (X) ! BEnd(A), in the 1-category Ind(Pro(S)). This construction provide us with a well defined, up to contractible indeterminacy,

p A : LocSys framed `,n (X)(A) ! Map
Ind(Pro(S)) B⇡ ét 1 (X), BEnd(Z) . On the other hand, the morphisms p A assemble to provide a morphism of stacks p : LocSys framed `,n (X) ! t 0 RLocSys `,n (X). Let Z 2 dAfd op Q `be a derived Q `-affinoid space. Let ⇢ 2 RLocSys `,n (X)(O Z ) be a continuous representation with values in O Z . The tangent complex of RLocSys `,n (X) at ⇢ is defined as the fiber

T RLocSys `,n (X),⇢ := fib ⇢ (p O Z )
where

p O Z : RLocSys `,n (X)(O Z an O Z ) ! RLocSys `,n (O Z ),
is the morphism of stacks induced from the canonical projection map

p O Z ,O Z : O Z O Z ! O Z .
The derived stack RLocSys `,n is not, in general, representable as derived Q `-analytic stack, as this would entail the representability of its 0-truncation. Nevertheless we can compute its tangent complex explicitly: Lemma 6.3.1.11. [1, Proposition 4.4.9.] Let ⇢ 2 RLocSys `,n (X)(O Z ). We have a natural morphism

T RLocSys `,n (X),⇢ ! C ⇤ ét (X, Ad (⇢)) [1],
which is an equivalence in the derived 1-category Mod (Z) .

Proof. The proof of [1, Proposition 4.4.9] applies.

The bounded ramification case

In this §we are going to define a natural derived enhancement of LocSys `,n, (X) and prove its representability by a derived Q `-analytic stack. Let X be a smooth scheme over an algebraically closed field k of positive characteristic p 6 = `. Definition 6.3.2.1. Consider the sub-site X tame ét of the small étale site X ét spanned by those étale coverings Y ! X satisfying condition (2) in Theorem 6.2.2.9. We can form the 1-topos Shv tame (X) := Shv (X tame ét ) of tamely ramified étale sheaves on the Grothendieck site X tame ét .

Consider the inclusion of sites ◆ : X tame ét ,! X ét , it induces a geometric morphism of 1-topoi

g ⇤ : Shv ét (X) ! Shv tame ét (X) (6.3.2.1)
which is a right adjoint functor to the functor induced by precomposition with ◆.

Lemma 6.3.2.2. The geometric morphism of 1-topoi g ⇤ : Shv tame ét (X) ! Shv ét (X) introduced in (6.3.2.1) is fully faithful.

Proof. As the Grothendieck topology on X tame ét is induced by the inclusion functor ◆ : X tame ét ! X ét , it suffices to prove the corresponding statement for the 1-categories of presheaves. More specifically, the statement of the lemma is a consequence of the assertion that the left adjoint

◆ ⇤ : PShv (X ét ) ! PShv X tame ét ,
given by precomposition along ◆, admits a fully faithful right adjoint. The existence of a right adjoint for ◆ ⇤ , denoted ◆ ⇤ , follows by the Adjoint functor theorem. The required right adjoint is moreover computed by means of a right Kan extension along ◆. Let Y 2 X tame ét , we can consider Y 2 X ét by means of the inclusion functor Proposition 6.3.2.9. Let q : ⇡ w 1 (X) ! be a surjective continuous group homomorphism whose target is finite. Then the 0-truncation of RLocSys `,n, (X) is naturally equivalent to LocSys `,n, (X). In particular, the former is representable by a Q `-analytic stack.

Proof. It suffices to prove the statement for the corresponding moduli associated to Sh ét (X), B⇡ w 1 (X) and B . Each of these three cases can be dealt as in Theorem 6.3.1.8.

Similarly to the derived moduli stack RLocSys `,n (X) we can compute the tangent complex of RLocSys `,n, (X) explicitly. In order to do so, we will first need some preparations: Construction 6.3.2.10. Let Y 2 Pro S fc 1 be a 1-connective profinite space. Fix moreover a morphism c : ⇤ ! X, in the 1-category Pro S fc . Notice that such choice is canonical up to contractible indeterminacy due to connectedness of X.

Let Perf (Q `) the 1-category of perfect Q `-modules. One can canonically enhance Perf(Q `) to an object in the 1-category ECat 1 of Ind(Pro(S))-enriched 1-categories. Consider the full subcategory

Perf `(Y ) := Fun cont (Y, Perf(Q `)) of Fun (Mat (Y ) , Perf(Q `)) spanned by those functors F : Y ! Perf(Q `) with M := F (⇤) such that the induced morphism ⌦Mat (X) ! End (M ) (6.3.2.2)
is equivalent to the materialization of a continuous morphism

⌦X ! End (M )
in the 1-category Ind(Pro(S)). Thanks to [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Corollary 4.3.23] the 1-category Perf `(X) is an idempotent complete stable Q `-linear 1-category which admits a symmetric monoidal structure given by point-wise tensor product.

Consider the ind-completion Mod Q `(X) := Ind (Perf `(X)), which is a presentable stable symmetric monoidal Q `-linear 1-category, [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Corollary 4.3.25]. We have a canonical functor p `(X) : Mod Q `(X) ! Mod Q `given informally by the formula

colim i F 2 Mod Q `(Y ) 7 ! colim i (F i (⇤)) 2 Mod Q `.
Given Z := (Z, O Z ) 2 dAfd Q `a derived Q `-affinoid space, we denote (Z) := (Z) the corresponding derived ring of global sections. Consider the extension of scalars 1-category

Mod (Z) (Y ) := Mod Q `(Y ) ⌦ Q ` (Z),
which is a presentable stable symmetric monoidal (Z)-linear 1-category, [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Corollary 4.3.25]. We can base change p `(Y ) to a well defined (up to contractible indeterminacy) functor p (Z) (Y ) : Mod (Z) (Y ) ! Mod (Z) given informally by the association

⇣ colim i F i ⌘ ⌦ Q ` (Z) 2 Mod (Z) (X) 7 ! colim i (F i (⇤) ⌦ Q ` (Z)) 2 Mod (Z) .
Proposition 6.3.2.11. Let Z 2 dAfd be a derived Q `-affinoid space and ⇢ 2 RLocSys `,n, (X)(O Z ). The inclusion morphism of stacks RLocSys `,n, (X) ,! RLocSys `,n (X) induces a natural morphism at the corresponding tangent complexes at ⇢ T RLocSys `,n, ,⇢ ! T RLocSys `,n ,⇢ is an equivalence in the 1-category Mod (Z) . In particular, we have an equivalence of (Z)-modules

T RLocSys `,n, , ⇢ ' C ⇤ ét (X, Ad (⇢)) [1] 2 Mod (Z) .
is a dualizable object of the derived 1-category Mod (Z) . Thanks to Theorem 6.3.2.13 we deduce that the existence of a cotangent complex is equivalent to the existence of a global cotangent complex for the derived moduli stack

RLocSys `,n Sh tame (Y U ) 2 dSt Afd Q `, ⌧ ét .
We are thus reduced to show that Sh tame (Y ) 2 Pro(S fc ) is cohomologically perfect and cohomologically compact, see [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Definition 4.2.7] and [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Definition 4.3.17] for the definitions of these notions. As Y U is a smooth scheme over a field of characteristic p 6 = `, cohomologically perfectness of Sh tame (Y U ) follows by finiteness of étale cohomology with `-adic coefficients, [START_REF] Milne | Lectures on étale cohomology[END_REF]Theorem 19.1] together with [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Proposition 3.1.7]. To show that Sh tame (Y ) is cohomologically compact we pick a torsion Z `-module N which can be written as a filtered colimit N ' colim ↵ N ↵ of perfect Z `-modules. As the tame fundamental group is topologically of finite type and for each i > 0, the stable homotopy groups ⇡ i Sh tame (Y U ) st are finitely presented the result follows. For these reasons, the derived moduli stack RLocSys `,n Sh tame (Y U ) admits a glocal cotangent complex. Theorem 6.3.2.13 implies now that the same is true for RLocSys `,n, (X). Compatibility with Postnikov towers of RLocSys `,n, (X) follows from the fact that the latter moduli is defined as a pullback of stacks compatible with Postnikov towers.

Comparison statements 6.4.1 Comparison with Mazur's deformation functor

Let L be a finite extension of Q `, O L its ring of integers and l := L /m L its residue field. We denote CAlg sm /l the 1-category of derived small k-algebras augmented over l.

Let G be a profinite group and ⇢ : G ! GL n (L) a continuous `-adic representation of G. Up to conjugation, ⇢ factors through GL n ( L ) ✓ GL n (L) and we can consider its corresponding residual continuous l-representation

⇢ : G ! GL n (l).
The representation ⇢ can the be obtained as the inverse limit of {⇢ n : G ! GL n (O L /m n+1 L )} n , where each ⇢ n ' ⇢ mod m n+1 . For each n 0, ⇢ n is a deformation of the residual representation ⇢ to the ring O L /m n+1 L . Therefore, in order to understand continuous representations ⇢ : G ! GL n (L) one might hope to understand residual representations ⇢ : G ! GL n (l) together with their corresponding deformation theory. For this reason, it is reasonable to consider the corresponding derived formal moduli problem, see [START_REF] Lurie | Spectral algebraic geometry[END_REF]Definition 12.1.3.1], associated to ⇢:

Def ⇢ : CAlg sm /l ! S,
given informally via the formula ] and its proof imply that one has an equivalence between the tangent complex of Def ⇢ and the complex of continuous cochains of Ad(⇢)

T Def⇢ ' C ⇤ cont (G, Ad(⇢)) [1] (6.4.1.2)
in the 1-category Mod l . Replacing BG in (6.4.1.1) by étale homotopy type of X, Sh ét (X), and C ⇤ cont by C ⇤ ét in (6.4.1.2) it follows by [START_REF] Milne | Lectures on étale cohomology[END_REF]Theorem 19.1] together with [21, Theorem 6.2.5] that Def ⇢ is pro-representable by a local Noetherian derived ring A ⇢ 2 CAlg /l whose residue field is equivalent to l. Moreover, A ⇢ is complete with respect to the augmentation ideal m A⇢ (defined as the kernel of the homomorphism ⇡ 0 (A ⇢ ) ! k of ordinary rings). It follows that A ⇢ admits a natural structure of a derived W (l)-algebra, where W (l) denotes the ring of Witt vector of l. As ⇢ admits deformations to O L , for e.g. ⇢ itself, we have that `6 = 0 in ⇡ 0 (A ⇢ ). By construction, the ordinary W (l)-algebra ⇡ 0 (A ⇢ ) pro-represents the functor t 0 (Def ⇢ ) : CAlg sm,/ l ! S. As a consequence, the mapping space on the right hand side of (6.4.1.3) is 0-truncated and the set of R-points corresponds to deformations of ⇢ valued in R. This is precisely Mazur's deformation functor, as introduced in [24, Section 1.2], concluding the proof.

Comparison with S. Galatius, A. Venkatesh derived deformation ring

In the case where X corresponds to the spectrum of a maximal unramified extension, outside a finite set S of primes, of a number field L and ⇢ : G X ! GL n (K) is a continuous representation, the corresponding derived W (k)-algebra was first introduced and extensively studied in [START_REF] Galatius | Derived Galois deformation rings[END_REF].

Comparison with G. Chenevier moduli of pseudo-representations

In this section we will compare our derived moduli stack RLocSys `,n (X) with the construction of the moduli of pseudo-representations introduced in [START_REF] Chenevier | The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings[END_REF]. We prove that RLocSys `,n (X) admits an admissible analytic substack which is a disjoint union of the various Def ⇢ . Such disjoint union of deformation functors admits a canonical map to the moduli of pseudo-representations of introduced in [START_REF] Chenevier | The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings[END_REF]. Such morphism of derived stacks is obtained as the composite of the 0-truncation functor followed by the morphism which associates to a continuous representation ⇢ its corresponding pseudo-representation, see [START_REF] Chenevier | The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings[END_REF]Definition 1.5]. Nevertheless, the derived moduli stack RLocSys `,n (X) has more points in general, and we will provide a typical example in order to illustrate this phenomena. Proposition 6.4.3.1. Let ⇢ : ⇡ ét 1 (X) ! GL n (F `) be a continuous residual `-adic representation. To ⇢ we can attach a derived Q `-analytic space Def rig ⇢ 2 dAn Q `for which every closed point ⇢ : Sp L ! Def rig ⇢ is equivalent to a continuous deformation of ⇢ over L.

Proof. Denote by dfSch W (l) the 1-category of derived formal schemes over W (l), introduced in [23, section 2.8]. The local Noetherian derived W (l)-algebra A ⇢ is complete with respect to its maximal ideal m A⇢ . For this reason, we can consider its associated derived formal scheme Spf A ⇢ 2 dfSch W (l) .

Let A 2 CAlg W (l) denote an admissible derived W (l)-algebra, see [2, Definition 3.1.1]. We have an equivalence of mapping spaces

Map dfSch W (l) (Spf A, Spf A ⇢ ) ' Map CAlg ad W (l) (A ⇢ , A) .
Notice that as A is a `-complete topological almost of finite type over W (k), the image of each t 2 m A⇢ is necessarily a topological nilpotent element of the ordinary commutative ring ⇡ 0 (A). Let m ✓ ⇡ 0 (A) denote a maximal ideal of ⇡ 0 (A) and let (A) m denote the m-completion of A. There exists a faithfully flat morphism of derived adic W (k)-algebra

A ! A 0 := Y m✓⇡0(A) (A) m
where the product is labeled by the set of maximal ideals of ⇡ 0 (A). By fppf descent we have an equivalence of mapping spaces

Map

CAlg ad

W (k) (A ⇢ , A) ' lim [n]2 op Map CAlg ad W (k) ⇣ A ⇢ , A 0
W (k) ⇣ A ⇢ , A 0 [n] ⌘ ' Def ⇢ ⇣ A 0 [n]
⌘ .

For each [n] 2 op we obtain thus a natural inclusion morphism ✓ [n] : Map

CAlg ad W (k) ⇣ A ⇢ , A 0 [n] ⌘ ! RLocSys `,n (X)(A 0 [n]
). The ✓ [n] assemble together and by fppf descent induce a morphism ✓ : Map

CAlg ad W (k) (A ⇢ , A) ! RLocSys `,n (X)(A).
By construction, ✓ induces a natural map of mapping spaces

Map CAlg ad W (k) (A ⇢ , A) ! Y m✓⇡0(A) ✓ RLocSys `,n (X)(A) ⇥ Def⇢(A m ) RLocSys `,n (X)(A m)
◆ which is equivalence of spaces. In order words Spf A ⇢ represents the moduli functor which assigns to each affine derived formal scheme Spf A, over W (l), the space of continuous representations ⇢ : Sh ét (X) ! BGL n (A) such that for each maximal ideal m ✓ ⇡ 0 (A) the induced representation

(⇢) m : Sh ét (X) ! BGL n ((A m))
is a deformation of ⇢ : Sh ét (X) ! BGL n (k).

f : Z ! (Spf A ⇢ )
rig in dAn Q `admits necessarily a formal model, i.e., it is equivalent to the rigidification of a morphism

f : Spf A ! Spf A ⇢ ,
where A 2 CAlg ad W (k) is a suitable admissible derived W (l)-algebra. The proof now follows from our previous discussion.

The proof of Theorem 6.4.3.1 provides us with a canonical morphism of derived moduli stacks Def rig ⇢ ! LocSys `,n (X). Therefore, passing to the colimit over all continuous representations

⇢ : ⇡ ét 1 (X) ! GL n (F `)
provides us with a morphism ✓ :

a ⇢ Def rig ⇢ ! RLocSys `,n (X) (6.4.3.2)
in the 1-category dSt(dAfd Q `, ⌧ ét ).

Proposition 6.4.3.2. The morphism of derived Q `-analytic stacks ✓ :

a ⇢ : ⇡ ét 1 (X)!GLn( Q`) Def rig ⇢ ! LocSys `,n (G)
displayed in (6.4.3.2) exhibits the left hand side as an analytic subdomain of the right hand side.

Proof. Let ⇢ : ⇡ ét 1 (X) ! GL n (F `) be a continuous representation. The induced morphism

✓ ⇢ : Def rig ⇢ ! RLocSys `,n (X)
is an étale morphism of derived stacks, which follows by noticing that ✓ ⇢ induces an equivalence at the level of tangent complexes. Moreover, Theorem 6.4.3.1 implies that ✓ ⇢ : Def rig ⇢ ! RLocSys `,n (X) exhibits the former as a substack of the latter. It then follows that the morphism is locally an admissible subdomain inclusion. The result now follows. Theorem 6.4.3.2 implies that RLocSys `,n (X) admits as an analytic subdomain the disjoint union of those derived Q `-analytic spaces Def rig ⇢ . One could then ask if ✓ is itself an epimorphism of stacks and thus an equivalence of such. However, this is not the case in general as the following example illustrates: Example 6.4.3.3. Let G = Z `with its additive structure and let A = Q `hT i be the (classical) Tate Q `-algebra on one generator. Consider the following continuous representation

⇢ : G ! GL 2 (Q `hT i),
given by

1 7 !  1 T 0 1 .
It follows that ⇢ is a Q `hT i-point of LocSys `,n (Z `) but it does not belongs to the image of the disjoint union Def rig ⇢ as ⇢ cannot be factored as a point belonging to the interior of the closed unit disk Sp (Q `hT i). Remark 6.4.3.4. As Theorem 6.4.3.3 suggests, when n = 2 the derived moduli stack RLocSys `,n (X) does admit more points than those that come from deformations of its closed points. However, we do not know if RLocSys `,n can be written as a disjoint union of the closures of Def rig ⇢ in LocSys `,n (X). However, when n = 1 the analytic subdomain morphism ✓ is an equivalence in the 1-category dSt dAfd Q `, ⌧ ét .

6.5 Shifted symplectic structure on RLocSys `,n (X)

Let X be a smooth and proper scheme over an algebraically closed field of positive characteristic p > 0. Poincaré duality provide us with a canonical map

' : C ⇤ ét (X, Q `) ⌦ Q `C ⇤ ét (X, Q `) ! Q `[ 2d] in the derived 1-category Mod Q `is non-degenerate, i.e., it induces an equivalence of derived Q `-modules C ⇤ ét (X, Q `) ! C ⇤ ét (X, Q `)_ [ 2d], (6.5.0.1)
in Mod Q `. As we have seen in the previous section, we can identify the left hand side of (6.5) with a (shit) of the tangent space of RLocSys `,n (X) at the trivial representation. Moreover, the equivalence holds if we consider étale (co)chains with more general coefficients. The case that interest us is taking étale cohomology with Ad(⇢)-coefficients for a continuous representation ⇢ : ⇡ ét 1 (X) ! GL n (A), with A 2 Afd Q `. Let ⇢ 2 RLocSys `,n (X)(Z), we can regard ⇢ as a dualizable object of the symmetric monoidal 1-category Perf ad `(X ) := Fun ECat1 Sh ét (X), Perf(A) . Let ⇢ _ denote a dual for ⇢. By definition of dualizable objects, we have a canonical trace map

tr ⇢ : ⇢ ⌦ ⇢ _ ! 1 Perf ad `(X )
in the 1-category Perf ad `(X ) and 1

Perf ad `(X ) denotes the unit object of the latter 1-category. Therefore, passing to mapping spaces, we obtain a natural composite

Map

Perf ad `(X ) (1, Ad(⇢)) ⌦ Map Perf ad `(X ) (1, Ad(⇢)) mult ! Map Perf ad `(X ) (1, Ad(⇢)) (6.5.0.2) tr⇢ ! Map Perf ad `(X ) (1, 1) (6.5.0.3)
in the 1-category Mod (Z) . By identifying the above with étale cohomology coefficients with coefficients we obtain a non-degenerate bilinear form

C ⇤ ét X, Ad(⇢) [1] ⌦ C ⇤ ét (X, Ad(⇢) [1] ! C ⇤ ét X, Ad(⇢) [2] tr⇢ ! C ⇤ ét X, (Z) [2 2d] (6.5.0.4)
in the 1-category Mod (Z) . Moreover, this non-degenerate bilinear form can be interpreted as a Poincaré duality statement with Ad(⇢)-coefficients.

Our goal in this §is to construct a shifted symplectic form ! on RLocSys `,n (X) in such a way that its underlying bilinear form coincides precisely with the composite (6.5.0.4). We will also analyze some of its consequences. Before continuing our treatment we will state a Q `-analytic version of the derived HKR theorem, first proved in the context of derived algebraica geometry in [START_REF] Toën | Algebres simpliciales S 1-équivariantes, théorie de de Rham et théoremes HKR multiplicatifs[END_REF]. Theorem 6.5.0.1 (Analytic HKR Theorem). Let k denote either the field of complex numbers or a non-archimedean field of characteristic 0 with a non-trivial valuation. Let X 2 dAn k be a derived k-analytic space. Then there is an equivalence of derived analytic spaces X ⇥ X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF], compatible with the projection to X.

The proof of Theorem 6.5.0.1 is a work in progress together with F. Petit and M. Porta, which the author hopes to include in his PhD thesis.

Shifted symplectic structures

In this §we fix X a smooth scheme over an algebraically closed field k of positive characteristic p.

In [START_REF] Toën | Structures symplectiques et de Poisson sur les champs en catégories[END_REF] the author proved the existence of shifted symplectic structures on certain derived algebraic stacks which cannot be presented as certain mapping stacks. As RLocSys `,n (X) cannot be presented as usual analytic mapping stack, we will need to apply the results of [START_REF] Toën | Structures symplectiques et de Poisson sur les champs en catégories[END_REF] to construct the desired shifted sympletic structure on RLocSys `,n (X). Definition 6.5.1.1. Consider the canonical inclusion functor ◆ : dSt (dAfd Q `, ⌧ ét , P sm ) ✓ Fun (dAfd Q `, S). The functor ◆ admits a left adjoint which we refer to as the stackification functor ( ) st : Fun (dAfd Q `, S) ! dSt (dAfd Q `, ⌧ ét , P sm ). Definition 6.5.1.2. Consider the functor PerfSys f `: dAfd Q `! S which is defined via the assignment

Z 2 dAfd op Q `7 ! Map ECat1 ⇣ Sh ét (X), Perf (Z) ⌘ 2 S
where we designate Perf (Z) to be the Ind(Pro(S))-enriched 1-category of perfect (Z)-modules, which is equivalent to the subcategory of dualizable objects in the 1-category of Tate modules on (Z), Mod Tate (Z) , [?]. We define the moduli stack PerfSys `2 dSt (dAfd Q `, ⌧ ét , ) as the stackyfication of PerfSys f `. Remark 6.5.1.3. This is an example of a moduli stack which cannot be presented as a usual mapping stack, instead one should think of it as an example of a continuous mapping stack. Notation 6.5.1.4. We will denote Cat ⌦ 1 the 1-category of (small) symmetric monoidal 1-categories. Definition 6.5.1.5. Let C 2 Cat ⌦ 1 be a symmetric monoidal 1-category. We say that C is a rigid symmetric monoidal 1-category if every object C 2 C is dualizable. Notation 6.5.1.6. We denote by Cat st,!,⌦ 1 the 1-category of small rigid symmetric monoidal 1-categories.

Consider the usual inclusion of 1-categories S ,! Cat 1 , it admits a right adjoint, denoted ( ) ' : Cat 1 ! S which we refer as the underlying 1-groupoid functor. Given C 2 Cat 1 its underlying 1-groupoid C ' 2 S consists of the maximal subgroupoid of C, i.e., the subcategory spanned by equivalences in C. Lemma 6.5.1.7. There exists a valued Cat st,!,⌦

1 -valued pre-sheaf Perf ad `(X ) : dAfd Q `! Cat 1
given on objects by the formula

Z 2 dAfd Q `7 ! Fun ECat1 X, Perf (Z) .
Moreover, the underlying derived stack ( ) ' Perf ad `(X ) 2 dSt Afd Q `, ⌧ ét is naturally equivalent to derived stack PerfSys `2 dSt (dAfd Q `, ⌧ ét ).

Proof. The construction of Perf ad `(X ) is already provided in [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Definition 4.3.11]. Moreover, it follows directly from the definitions that Perf ad `(X ) ' ' PerfSys `(X ).

Theorem 6.5.1.7 is useful because it place us in the situation of [32, §3]. Therefore, we can run the main argument presented in [32, §3]. Before doing so, we will need to introduce some more ingredients: Definition 6.5.1.8. Let H

⇣

Perf ad `(X ) ⌘ : dAfd op Q `! S denotes the sheaf defined on objects via the formula

Z 2 dAfd op Q `7 ! Map Perf ad `(X ) (Z) (1, 1) 2 S,
where 1 2 Perf ad `(X )( (Z)) denotes the unit of the corresponding symmetric monoidal structure on Perf ad `( (Z)). Definition 6.5.1.9. Let O : dAfd op Q `! CAlg Q `denote the sheaf on (Afd Q `, ⌧ ét ) given on objects by the formula

Z 2 dAfd op Q `7 ! (Z) 2 CAlg Q `.
Construction 6.5.1.10. One is able to define a pre-orientation, in the sense of [START_REF] Toën | Structures symplectiques et de Poisson sur les champs en catégories[END_REF]Definition 3.3], on the Cat st,!,⌦

1 -value stack Perf ad `(X ) ✓ : H ⇣ Perf ad `(X ) ⌘ ! O[ 2d],
as follows: let Z 2 dAfd Q `be a derived Q `-affinoid space. We have a canonical equivalence in the 1-category

Mod (Z) (Z) : Map Perf ad `(X )( (Z)) (1, 1) ' C ⇤ ét (X, (Z)) , (6.5.1.1)
by the very construction of Perf ad ` (Z) . Moreover, the projection formula for étale cohomology produces a canonical equivalence

C ⇤ et (X, (Z)) ' C ⇤ et (X, Q `) ⌦ Q ` (Z) in the 1-category Mod Q `.
As X is a connected smooth scheme of dimension d over an algebraically closed field we have a canonical map on cohomology groups

↵ : Q `' H 0 (X ét , Q `) ⌦ H 2d (X ét , Q `) ! Q ẁhich
is induced by Poincaré duality. Consequently, the morphism ↵ induces, up to contractible indeterminacy, a canonical morphism

C ⇤ ét (X, Q `) ! Q `[ 2d]. (6
✓ : H ⇣ Perf ad `(X ) ⌘ ! O[ 2d],
which corresponds to the desired orientation. Given Z 2 dAfd Q `, the 1-category Perf ad ` (Z) is rigid. Thus for a given object ⇢ 2 Perf ad ` (Z) we have a canonical trace map tr ⇢ : Ad (⇢) ! 1. which together with the symmetric monoidal structure provide us with a composite of the form

Map

Perf ad `(X )( (Z)) (1, Ad(⇢)) ⌦ Map Perf ad `(X )( (Z)) (1, Ad(⇢)) !Map Perf ad `(X )( (Z)) (1, Ad(⇢) ⌦ Ad(⇢)) (6.5.1.3) ! Map Perf ad `(X )( (Z)) (1, Ad(⇢)) !Map Perf ad `(X )( (Z) (1, 1) ! (Z)[ 2d] (6.5.1.4)
which we can right equivalently as a morphism

C ⇤ ét (X, Ad(⇢)) ⌦ C ⇤ ét (X, Ad(⇢)) ! (Z)[2 2d],
which by our construction coincides with the base change along Q `! (Z) of the usual pairing given by Poincaré Duality. Lemma 6.5.1.11. Let Z 2 dAfd Q `be a derived Q `-affinoid space. The pairing of Theorem 6.5.1.10

Map

Perf ad

`(X )( (Z) (1, Ad(⇢)) ⌦ Map Perf ad `(X )( (Z)) (1, Ad(⇢)) ! (Z)[ 2d]
is non-degenerate. In particular, the pre-orientation ✓ :

H ⇣ Perf ad `(X ) ⌘ ! O[ 2d]
is an orientation, see [START_REF] Toën | Structures symplectiques et de Poisson sur les champs en catégories[END_REF]Definition 3.4] for the latter notion.

Proof. Let ⇢ 2 PerfSys `(X )(O Z ) be an arbitrary continuous representation with O Z -coefficients. We wish to prove that the natural mapping

Map

Perf ad `(X )( (Z) (1, Ad(⇢)) ⌦ Map Perf ad `(X )( (Z)) (1, Ad(⇢)) ! (Z)[ 2d]
is non-degenerate. As Z lives over Q `and p 6 = `it follows that ⇢ 2 PerfSys `, (X) for a sufficiently large finite quotient q : ⇡ w 1 (X) ! . It then follows by [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]Proposition 4.3.19] together with Theorem 6.3.2.13 that ⇢ can be realized as the B -fixed points of a given e ⇢ : Sh tame (Y ) ! BGL n (A 0 ), where Y ! X is a suitable étale covering and A 0 2 CAlg ad k is an admissible derived Z `-algebra such that

Spf A 0 rig ' Z,
in the 1-category dAfd Q `. We notice that it suffices then to show the statement for the residual representation

⇢ 0 : Sh tame (Y ) ! BGL n (A 0 /`)
, where A 0 /`denotes the pushout

A 0 [t] A 0 A 0 A 0 /t 7 !t 7 !0
computed in the 1-category CAlg ad k . We can write A 0 /`as a filtered colimit of free F `-algebras F `[T 0 , . . . , T m ], where the T i sit in homological degree 0. As Sh tame (Y ) is cohomological compact we reduce ourselves to prove the statement by replacing ⇢ 0 with a continuous representation with values in some polynomial algebra F `[T 0 , . . . , T m ]. The latter is a flat module over F `. Therefore, thanks to Lazard's theorem [START_REF] Lurie | Higher algebra[END_REF]Theorem 8.2.2.15] we can further reduce ourselves to the case where ⇢ 0 is valued in a finite F `-module. The result now follows by the Theorem 6.5.1.10 together with the projection formula for étale cohomology and Poincaré duality for étale cohomology.

As a corollary of [START_REF] Toën | Structures symplectiques et de Poisson sur les champs en catégories[END_REF]Theorem 3.7] one obtains the following important result: Theorem 6.5.1.12. The derived moduli stack PerfSys `(X ) 2 dSt (dAfd Q `, ⌧ ét ) admits a canonical shifted symplectic structure ! 2 HC PerfSys `(X ) , where the latter denotes cyclic homology of the derived moduli stack PerfSys `(X ). Moreover, given Z 2 dAfd Q `and ⇢ 2 PerfSys ` (Z) , the shifted symplectic structure ! on PerfSys `(X ) is induced by étale Poincaré duality

C ⇤ ét (X, Ad(⇢)) [1] ⌦ C ⇤ ét (X, Ad(⇢)) [1] ! (Z)[2 2d].
Proof. This is a direct consequence of our previous discussion together with the argument used in [32, Theorem 3.7].

Applications

Consider the canonical inclusion ◆ : RLocSys `,n (X) ,! PerfSys `(X ). Pullback along the morphism ◆ on cyclic homology induces a well defined, up to contractible indeterminacy, morphism ◆ ⇤ : HC PerfSys `(X ) ! HC RLocSys `,n (X) .

We then obtain a canonical closed form ◆ ⇤ (!) 2 HC RLocSys `,n (X) . Moreover, as ◆ induces an equivalence on tangent complexes, the closed form ◆ ⇤ (!) 2 HC RLocSys `,n (X) is non-degenerate, thus a 2 2d-shifted symplectic form. Similarly, given a finite quotient q : ⇡ w 1 (X) ! , we obtain a 2 2d-shifted symplectic form on the derived Q `-analytic stack RLocSys `,n, (X). The existence of the sifted symplectic form entails the following interesting result: as a morphism

HH BGL an n ⌦ C ⇤ ét (X, Q `)_ ! HH RLocSys `,n (X) in the 1-category Mod Q `.
The analytic HKR theorem then provide us with the desired morphism

C ⇤ dR BGL an n ⌦ C ⇤ ét X, Q ` _ ! C ⇤ dR RLocSys `,n (X)
in the 1-category Mod Q `.

C ⇤ ét BGL n , Q ` ⌦ C ⇤ ét X, Q ` ! C ⇤ dR RLocSys `,n (X) .
in the 1-category Mod Q `. As C ⇤ dR RLocSys `,n (X) admits a natural E 1 -algebra structure we obtain, by the universal property of the Sym construction, a well defined morphism

Sym C ⇤ ét BGL n , Q ` ⌦ C ⇤ ét X, Q ` ! C ⇤ dR RLocSys `,n (X) . (6.5.2.3) in the 1-category CAlg Q `.
Assuming further that X is a proper and smooth curve over an algebraically closed field, an `-adic version of Atiyah-Bott theorem proved in [START_REF] Gaitsgory | Weil's conjecture for function fields[END_REF] implies that we can identify the left hand side of (6.5.2.3) with a morphism

C ⇤ ét (Bun GLn (X), Q `) ! C ⇤ dR RLocSys `,n (X) in the 1-category CAlg Q `.
As a corollary we obtain: Corollary 6.5.2.5. Let X be a smooth scheme over an algebraically closed field of positive characteristic p > 0. We have a canonical morphism

' : C ⇤ ét BGL n , Q ` ⌦ C ⇤ ét X, Q ` _ ! C ⇤ dR RLocSys `,n (X)
in the 1-category CAlg Q `. Moreover, assuming further that X is also a proper curve we obtain a canonical morphism

C ⇤ ét Bun GLn X , Q ` ! C ⇤ dR RLocSys `,n (X) in the 1-category CAlg Q `.
Remark 6.5.2.6. By forgetting the mixed k-algebra structure on C ⇤ dR RLocSys `,n (X) one can prove that the moprhism ' sends the product of the canonical classes on

C ⇤ ét BGL n , Q ` ⌦ C ⇤ ét (X, Q `)_
to the underlying cohomology class of the shifted symplectic form ! on RLocSys `,n (X).

Step 2: Passage to nonconnective objects We can describe the most basic idea behind our strategy as follows: instead of proving directly the equivalence between the monads T S 1 and T " , we look at the comonads. Indeed, the functor U S 1 : S 1 -A ! A also admits a right adjoint, given by sending A 2 A to

A S 1 := A ⇥ A⇥A A.
In the cases of our interest, it turns out that the forgetful functor U " : "-A ! A also admits a right adjoint, given by sending A 2 A to the split square-zero extension A ⌧ 0 (A [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). This implies that both S 1 -A and "-A are comonadic over A. Let C S 1 and C " denote the respective comonads. It is then enough to prove that C S 1 and C " are equivalent comonads, but unfortunately this comparison is as difficult as the original problemma.

In order to decrease the complexity of the problemma, we would like to prove that our comonads to be induced by coalgebra objects in A, via the correspondence provided by the following lemmama: Lemma 7.1.0.7 (cf. ??). Let C ⌦ be a symmetric monoidal 1-category. The functor

C ! End(C)
informally given by X 7 ! X ⌦ induces a well defined functor

CoAlg(C ⌦ ) ! CoMonads(C).
We refer to the comonads in the essential image of this functor as representable comonads.

Indeed, if could construct two coalgebras A S 1 and A " in A whose associated comonads are C S 1 and C " , it would then be enough to prove that A S 1 and A " are equivalent in the 1-category CoAlg(A). Unfortunately, this does not happen in our case. Nevertheless, we can make it true after suitably enlarging the 1-category A.

Example 7.1.0.8. In order to get a feeling for what is the obstruction to exhibit C S 1 as the comonad associated to a coalgebra A S 1 , it is useful to look at the algebraic situation. In this case, the 1-category A coincides with the 1-category of simplicial commutative algebras CAlg k . The comonad C S 1 sends an object A 2 CAlg k to A ⇥ A⇥A A. On the other hand, if R 2 CoAlg(CAlg k ), then its associated comonad sends A to A ⌦ R. If we assume that C S 1 (A) can be written as A ⌦ R for every choice of A, then we would obtain

R ' k ⌦ R ' C S 1 (k) ' k ⇥ k⇥k k.
But in CAlg k one has k ⇥ k⇥k k ' k, and the comonad associated to k is simply the identity. On the other hand, when A is not discrete, then the underlying module of A ⇥ A⇥A A is not equivalent to A itself, but rather to

A ⌧ 0 (A[ 1]).
This is suggesting that working with connective commutative algebras is too much restrictive for this problemma. Using the Dold-Kan equivalence we can identify CAlg k with the underlying 1-category cdga 0 k of connective cdgas (we use homological convention). The inclusion cdga 0 k ,! cdga k does not commute with limits, and in cdga k one has k⇥ k⇥k k ' k k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. Notice that all the functors introduced so far (U S 1 , S 1 ⌦ , DR, . . . ) extend to the unbounded setting. Furthermore, in the unbounded setting we always have

C S 1 (A) ' A ⌦ (k ⇥ k⇥k k).
In the light of the above example, our actual strategy can be summarized as follows:

(i) Construct a "nonconnective enlargement" A nc of the 1-category A.

(ii) Prove that the comonads C S 1 and C " extend to comonads C nc S 1 and C nc " in A nc .

(iii) Prove that the extended comonads C nc S 1 and C nc " are representable by objects A S 1 and A " in CoAlg(A nc ).

(iv) Finally prove that A S 1 and A " are equivalent in CoAlg(A nc ).

The construction of the nonconnective enlargement A nc in the analytic setting constitutes the technical heart of this paper. The ideas introduced are a natural extension of the ones that appear in [Lur11d], and they allow to construct nonconnective enlargements in many situations. We believe that the 1-categories of nonconnective structures that we construct here as an auxiliary tool to prove the HKR theorem are interesting on their own and will prove useful in a variety of different situations.

Let us now give more details about the important features that a "nonconnective enlargement" A nc should satisfy in order to be useful to our problemma. Assumption 7.1.0.9. There exists a presentable 1-category A nc equipped with a connective cover functor

⌧ 0 : A nc ! A.
Furthermore, this functor admits a fully faithful left adjoint

i : A , ! A nc .
This assumption together with the chain rule for Goodwillie's derivative implies that for every A 2 A there is a fully faithful functor @(i) : A-Mod ! Sp(A nc /i(A) ). In turn, this allows to prove that for A 2 A the cotangent complex L A coincides with the cotangent complex L i(A) of i(A) 2 A nc . It also allows to define nonconnective split square-zero extensions: if A 2 A nc and M 2 Sp(A nc /A ), we set

A nc M := ⌦ 1 (M ),
where ⌦ 1 is the natural functor ⌦

1 : Sp(A nc /A ) ! A nc /A .
Assumption 7.1.0.10. There exist coalgebras A " , A S 1 2 CoAlg(A nc ) such that for every A 2 A one has a canonical equivalences

i(A) ⌦ A " ' i(A) nc @(i)(A[ 1]) and i(A) ⌦ A S 1 ' i(A) ⇥ i(A)⇥i(A) i(A).
This assumption is easily verified when A nc = cdga k . In the analytic setting however its verification is one of the most delicate points of the paper. It requires a relative version of the Van Est theorem, which we will discuss in the next step. However, once proven it implies that "-A and S 1 -A can be identified with the full subcategories of CoMod A" (A nc

) and CoMod A S 1 (A nc ) spanned by connective objects. In other words, we have the following pullback diagrams

"-A CoMod A" (A nc ) A A nc U" i , S 1 -A CoMod A S 1 (A nc ) A A nc . U S 1 i
In this way, we are reduced to prove that there is an equivalence in CoAlg(A nc ) between A S 1 and A " . In order to do this, we need one final structural property of the 1-category A nc : Assumption 7.1.0.11. There exists a conservative functor

U : A nc ! Mod k ,
where Mod k denotes the 1-category of (unbounded) k-modules. Furthermore, this functor admits a left adjoint, denoted Sym nc k .

Step 3: Use of a theorem of type Van Est Using the conservativity of the functor U : A nc ! Mod k , we see that it would be enough to construct a morphism A S 1 ! A " in CoAlg(A nc ) that becomes an equivalence in Mod k . Unfortunately, it is unreasonable to be able to construct such a morphism directly. This can be seen by attempting to work over the sphere spectrum k = S instead of over a field of characteristic 0.

Example 7.1.0.12. Let k = S be the sphere spectrum. Then Böckstedt's computation shows that

⇡ ⇤ THH(HF p ) = F p [ ],
with | | = 2. On the other hand, ⇡ 2 (L HFp/S ) = F p , which implies that ⇡ 3 (Sym HFp (L HFp/S [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF])) = F p . Therefore THH(HF p ) = HF p ⌦ HFp⌦ S HFp HF p cannot be equivalent to Sym HFp (L HFp/S [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]), and so even the plain HKR theorem fails over the sphere spectrum. On the other hand, A S 1 is S ⇥ S⇥S S and A " is the split square-zero extension S S [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. In particular, the underlying spectra of A S 1 and A " are equivalent. This means that the equivalence of the underlying modules cannot even be lifted to an equivalence at the E 1 level, let alone at the bialgebra level.

This example suggests that it is more reasonable to look for a correspondence between A S 1 and A " . When working over a field of characteristic zero, both morphisms in this correspondence will be equivalences, but in general they will not. Notice that

U (A S 1 ) is forced to be k ⇥ k⇥k k ' k k[ 1]. Assumption 7.1.0.13. One has U (A " ) = k k[ 1].
This assumption is easy to verify in all cases of interests. When A nc = cdga it is a consequence of [Lur12c,7.3.4.15]. In the analytic case, it is easy to reduce oneself to the algebraic situation. This provides us with the following canonical correspondence in A nc :

Sym nc k (k[ 1]) A S 1 A " .
p q Assumption 7.1.0.14. Both U (p) and U (q) are equivalences in Mod k .

We warn the reader that the above assumption is really strong and it is not always satisfied. For instance, when k has characteristic p > 0 one has

⇡ 1 (U (Sym nc k (k[ 1]))) = M N k.
In practice, in order to verify this assumption we need to really unravel the construction of A nc and of the functor Sym nc k . The reason we are able to go through this computation is that A nc is constructed in a fairly geometric way, and U (Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF])) can be identified with the cohomology complex of the classifying stack B(BG a ), where BG a denotes the analytic 1-dimensional additive group. To actually compute this cohomology, we resort to a theorem due to Van Est, that identifies the group cohomology of BG a with the Lie algebra cohomology of its Lie algebra.

Warning 7.1.0.15. The above discussion is simplistic. It is only accurate when k = C or it is a non-archimedean field of equicharacteristic 0. In the mixed characteristic case one needs to replace the middle comparison term with the nonconnective analytic algebra of global sections of the classifying stack B(D 1 k (r)), where D 1 k (r) denotes the non-archimedean closed disk of radius r, and r is supposed to be less or equal the converging radius of the exponential. In this case, the theorem of Van Est is replaced by its non-archimedean analogue, which is due to Lazard.

underyling derivation associated to the mixed algebra appear naturally. Secondly, it generalizes to the analytic setting.

In Section 7.2.2 we turn our attention to S 1 -algebras. As before, we provide two equivalent description of this 1-category, one that is naturally monadic and one that is comonadic. The equivalence between these two description builds on the presentation as Segal objects of modules for a monoid in a Cartesian 1-category. This comparison allows us to prove the equivalence between mixed algebras and S 1 -algebras by checking that the comonads are equivalent.

Finally, in Section 7.2.3 we realize the general strategy described in the introduction by proving the HKR statement in the algebraic setting. As main auxiliary step, we prove the contractibility of the space of coalgebra structures on Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]).

Mixed algebras

Let C ⌦ be a symmetric monoidal 1-category. We set

CAlg(C) := Alg E1 (C ⌦
).

Given

A monadic presentation for mixed algebras

Fix a field k of characteristic 0. We let Mod k denote the stable 1-category of (unbounded) k-modules and Perf k be the full stable subcategory spanned by perfect complexes. We endow Mod k with its canonical symmetric monoidal structure. We set CAlg nc k := CAlg(Mod k ). It can be identified with the 1-category of unbounded cdgas. Given A 2 CAlg nc k we have a canonical equivalence

A-Mod ' Sp(CAlg nc k/A ).
For M 2 A-Mod we set

A M := ⌦ 1 (M ). In particular, we pose k["] := k k[1].
As an algebra, it coincides with the split square-zero extension of k by k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. Since k["] and all its finite tensor powers are formal, we can define a coalgebra structure on k["] simply by setting

(") := 1 ⌦ " + " ⌦ 1, where : k["] ! k["] ⌦ k k["]
denotes the comultiplication. It is easily checked that k["] inherits in this way the structure of a bialgebra.

A comonadic presentation for mixed algebras

We now provide a second construction for the 1-category of mixed algebras. We start by observing that the complex underlying k["] is perfect. As Perf k is a rigid symmetric monoidal 1-category, we deduce that the k-linear dual k[⌘] := Hom(k["], k) also acquires the structure of a bialgebra. In particular, k[⌘] is a coalgebra in Mod k . We let

⌘-Mod k := k[⌘]-CoMod(Mod k ) denote the 1-category of k[⌘]-comodules. Write u ⌘ : ⌘-Mod k ! Mod k
for the canonical forgetful functor.

Since k[⌘] is a bialgebra, we can also consider the 1-category

⌘-CAlg nc k := k[⌘]-CoMod(CAlg nc k ).
We let

v ⌘ : ⌘-CAlg nc k ! CAlg nc k
denote the natural forgetful functor. By construction, we obtain a commutative diagram

⌘-CAlg nc k CAlg nc k ⌘-Mod k Mod k . v⌘ U⌘ U u⌘
Moreover, the functors u ⌘ and v ⌘ are comonadic. Using ?? we can identify the respective comonads with the ones induced by the coalgebra structure on k[⌘] (considered as an elemmaent in Mod k , resp. in CAlg nc k ). In the next section we will prove that there is a canonical equivalence "-CAlg k ' ⌘-CAlg k . Before arguing about this, however, let us explore some of the basic features of the 1-category ⌘-CAlg nc k . We start by observing that if A 2 ⌘-CAlg nc k in particular we have a commutative triangle

A A ⌦ k k[⌘]
A,

⇤ : CAlg nc A//A ⌧ CAlg nc B//B : f ⇤ . From an informal point of view, f ⇤ sends an augmented A-algebra A ! R ! A to B ! B ⌦ A R ! B. Similarly, f ⇤ sends an augmented B-algebra B ! R ! B to A ! A ⇥ B R ! A. Lemma 7.2.1.5. Let f : A ! B be a morphism in CAlg nc k . The diagram B-Mod A-Mod CAlg nc B//B CAlg nc A//A ⌦ 1 f⇤ ⌦ 1 f⇤
is commutative and horizontally left adjointable.

which is determined by the condition that the underlying map A ! DR(A) coincides with the canonical inclusion, and the derivation L A/k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] ! DR(A) corresponds to the inclusion of L A/k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] in Sym k (L A/k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). This transformation induces a morphism of fiber sequences

Map

CAlg nc A (DR(A), B) Map CAlg nc k (DR(A), B) Map CAlg nc k (A, B) Der k (A, f ⇤ (B)) Map CAlg nc k (A, B ⌦ k k[⌘]) Map CAlg nc k (A, B).
As the outer vertical morphisms are equivalences and this holds for every choice of f : A ! B, we conclude that the morphism (7.2.1.1) induces a functorial equivalence

Map CAlg nc k (DR(A), B) ' Map CAlg nc k (A, B ⌦ k k[⌘]
), thus completing the proof of the claim.

Equivalence of the 1-categories of mixed algebras

In the previous sections we introduced two 1-categories, "-CAlg k and ⌘-CAlg k . We now prove that they are equivalent: Theorem 7.2.1.7. There is an equivalence :

"-CAlg k ' ⌘-CAlg k making the diagram "-CAlg nc k ⌘-CAlg nc k CAlg nc k v" ⇠ v⌘ commutative.
Proof. We observe that v ⌘ is comonadic by construction. ?? implies that the comonad associated to v ⌘ coincides with (k[⌘]), which is the comonad associated to the coalgebra k[⌘] 2 CoAlg(CAlg k ). On the other hand, Theorem 7.2.1.3 implies that the functor v " : "-CAlg nc k ! CAlg nc k is both monadic and comonadic. It is therefore enough to prove that the comonad associated to v " can be identified with (k[⌘]). Let us temporarily denote by C " the comonad associated to v " . Recall from ?? that we have a fully faithful functor

0 : CoAlg(CAlg nc k ) ! CoMonads lax (CAlg nc k )
. This functor has a right adjoint 0 , that is informally given by evaluation on k. Notice that C " can be naturally promoted to an elemmaent in CoMonads lax (CAlg nc k ). In particular, it is enough to construct an equivalence

↵ : 0 (k[⌘]) ! C " in CoMonads lax (CAlg nc k ). As the functor CoMonads lax (CAlg nc k ) ! End(CAlg nc k )
is conservative, it is enough to construct the morphism ↵ and to check afterwards that it induces an equivalence on the underlying endofunctors.

Since 0 is right adjoint to 0 , to produce a morphism 0

(k[⌘]) ! C " is equivalent to produce a morphism k[⌘] ! 0 (C " ) in CoAlg(CAlg k ).
We are therefore lead to identify 0 (C " ). Consider the commutative diagram where we denoted by ⇢ " and " the right adjoints to u " and v " , respectively. In other words, the diagram (7.2.1.2) is horizontally right adjointable. This provides us with the following alternative description for " : as u " is strong monoidal, the functor ⇢ " is lax monoidal. In particular, it induces a well defined functor CAlg nc k ! "-CAlg nc k , which coincides with " . Observe on the other hand that the coalgebra structure on k[⌘] corresponds to the algebra structure on k["]. Coupling this with the fact that k[⌘] is dualizable as k-module, we obtain a natural equivalence

"-CAlg nc k CAlg nc k "-Mod k Mod k . v" U" U u" (7
"-Mod k ' ⌘-Mod k ,
compatible with the forgetful functors to Mod k . This implies that the endofunctor u " ⇢ " is canonically identified with ⌦ k k[⌘]. As a consequence, we see that can identify 0 (C " ) with C " (k) ' k[⌘] with its canonical bialgebra structure. In other words, k[⌘] ' 0 (C " ). This provides us with the natural transformation ↵ : 0 (k[⌘]) ! C " we were looking for. To complete the proof, it is enough to observe that the previous discussion also showed that the endofunctor underlying C " conicides with ⌦ k k[⌘]. Therefore ↵ is an equivalence, and the proof is complete.

The equivalence provided by the previous theorem has the following two non-trivial consequences: Proof. The functor v " : "-CAlg nc k ! CAlg nc k is obviously monadic, hence the conclusion follows from Theorem 7.2.1.7.

Corollary 7.2.1.9. The endofunctor underlying the monadic functor u " : "-CAlg nc k ! CAlg nc k coincides with the functor

DR : CAlg nc k ! CAlg nc k informally sending A to Sym A (L A [1]).
Proof. We start with a simple consideration. Let U : C ! D be a functor between 1-categories. Suppose that U is both monadic and comonadic and let L (resp. R) denote its left (resp. right) adjoint. Then U L is left adjoint to U R.

Applying this remark to the forgetful functor u " : CAlg nc k ! CAlg nc k , we can characterize the endofunctor underlying the associated monad with the left adjoint to the endofunctor underlying the associated comonad. Using the equivalence provided by Theorem 7.2.1.7, we identify the latter with ⌦ k k[⌘]. At this point, the conclusion follows from Theorem 7.2.1.6.

Building on Theorem 7.2.2.4 it is easy to bootstrap and extend to the case of presheaves. Let us introduce the necessary terminology. Let C be an 1-category. Precomposition with the canonical functor ⇡ : C ! ⇤ gives rise to a functor

⇡ p : S ! PSh(C)
that sends a space K to the constant presheaf K on C associated to K. As ⇡ p commutes with limits, it can be promoted to a symmetric monoidal functor

⇡ p : S ⇥ ! PSh(C) ⇥ .
The E 1 -structure on S1 induces therefore a canonical E 1 -structure on S 1 For this, we will need to make certain assumptions on C itself.

To set the stage, suppose that C is a presentable 1-category. In this case C is canonically enriched with tensor and cotensor over S fin . In particular, we have a functor

⌦ : S fin ⇥ C ! C.
For any X 2 C we obtain an adjunction

⌦ X : S ⌧ C : Map C (X, ).
The right adjoint commutes with products and therefore it can be promoted to a symmetric monoidal functor (with respect to the cartesian monoidal structures on both sides). In particular, it can be canonically lifted to a functor

M X : Mon gp E1 (C ⇥ ) ! Mon gp E1 (S ⇥ ) fitting in the commutative diagram Mon gp E1 (S ⇥ ) Mon gp E1 (C ⇥ ) S C, M X Map C (X, ) (7.2

.2.4)

where the vertical arrows are the forgetful functors. Since they are conservative and commute with limits and sifted colimits, we deduce that M X admits a left adjoint, that we denote L X . When X = 1 C , we write L instead of L 1 C . We let and therefore we have to produce a natural isomorphism between L and L 0 . In order to do this, it is enough to prove that for every X 2 Mon gp E1 (S fin,⇥ ) there is a morphism

S 1 C := L(S 1 ) 2
⌘ : X ! M 1 C (L 0 (X)) in Mon gp E1 (S fin,⇥ ) inducing an equivalence Map Mon gp E 1 (S fin,⇥ ) (X, M 1 C (Y )) ' Map Mon gp E 1 (C ⇥ ) (L 0 (X), Y ). (7.2.2.5)
Represent X as a Segal object F X : op ! S fin . Since C satisfies the condition (M), we see that the functor

Map

C (1 C , F X ( ) ⌦ 1 C ) : op ! S fin still satisfies the Segal condition, and it corresponds precisely to M 1 C (L 0 (X)). The unit of the adjunction ⌦ 1 C : S C : Map C (1 C , ) induces therefore a natural transformation from F X to M 1 C (L 0 (X)), which is easily checked to induce an equivalence (7.2.2.5).

Since the Yoneda embedding y : C ! PSh(C) commutes with products, we see that y(S 1 C ) inherits the structure of a grouplike E 1 -monoid in PSh(C). Recall that the canonical functor ⇡ : C ! ⇤ induces an adjunction

⇡ p : S ⌧ PSh(C) : p ⇡,
and that furthermore ⇡ p commutes with all limits. In particular, this adjunction lifts to another adjunction

⇡ p : Mon gp E1 (S) ⌧ Mon gp E1 (PSh(C)) : p ⇡. Observe that p ⇡(y(S 1 C )) ' Map(1 C , S 1
C ). In particular, the unit of the adjunction L a M 1 C induces a morphism in Mon gp E1 (S)

S 1 ! p ⇡(y(S 1 C )).
In turn, this corresponds to a morphism in Mon gp E1 (PSh(C))

⇢ : S 1 = ⇡ p (S 1 ) ! y(S 1 C ),
which induces a forgetful functor

⇢ ⇤ : LMod y(S 1 C ) (PSh(C)) ! LMod S 1 (PSh(C)
). We would like to say that ⇢ ⇤ is an equivalence of 1-categories, but this will not be true in general. Therefore, we need to formulate some stronger assumption on C: Definition 7.2.2.8. Let C be a presentable 1-category. Let i : C 0 ✓ C be a full subcategory closed under products. We say that C 0 satisfies the condition (UM) (relative to C) if for every X 2 C 0 and every K 2 S fin the natural morphism

K ⌦ X ! (K ⌦ 1 C ) ⇥ X is an equivalence.
Remark 7.2.2.9. Notice that since C 0 is closed under products in C, the final object 1 C belongs to C 0 . In particular, if C 0 satisfies the condition (UM) and for any H 2 S fin the object H ⌦ 1 C still belongs to C 0 , then the natural morphism

K ⌦ (H ⌦ 1 C ) ! (K ⌦ 1 C ) ⇥ (H ⌦ 1 C ) is an equivalence. On the other hand, K ⌦ (H ⌦ 1 C ) ' (K ⇥ H) ⌦ 1 C . Map(K ⌦ (H ⌦ 1 C ), Y ) ' Map(K, Map(H ⌦ 1 C , Y )) ' Map(K, Map(H, Map(1 C , Y ))) ' Map(K ⇥ H, Map(1 C , Y ))
and therefore the Yoneda lemmama implies that K

⌦ (H ⌦ 1 C ) ' (K ⇥ H) ⌦ 1 C
. Therefore, we conclude that in this case C satisfies the condition (M). In particular, if C 0 = C satisfies (UM) then C satisfies (M) as well. However, in our applications C will be the opposite of the 1-category of nonconnective structures, and C 0 will be the full subcategory spanned by the connective ones. In this case, C satisfies the condition (M) and C 0 satisfies the condition (UM), but C 0 is not closed in C under tensor with finite spaces.

At this point, we are ready to state the main result of this section: Theorem 7.2.2.10. Let C be a presentable 1-category and let C 0 be a full subcategory of C closed under products. Suppose that C 0 satisfies the condition (UM). Then the functor ⇢

⇤ : LMod y(S 1 C ) (PSh(C)) ! LMod S 1 (PSh(C)) restricts to an equivalence of 1-categories LMod y(S 1 C ) (PSh(C)) ⇥ PSh(C) C 0 ' LMod S 1 (PSh(C)) ⇥ PSh(C) C 0 .
Proof. To simplify the notations, write

LMod y(S 1 C ) (C 0 ) := LMod y(S 1 C ) (PSh(C)) ⇥ PSh(C) C 0 and similarly Mod S 1 (C 0 ) := LMod S 1 (PSh(C)) ⇥ PSh(C) C 0 .
We start by observing that if X, Y 2 C 0 then the natural morphism

Map C ((S 1 C ) ⇥n ⇥ X, Y ) ! Map PSh(C) ((S 1 ) ⇥n ⇥ y(X), y(Y )) (7.2.2.6)
is an equivalence. Indeed, we can identify the right hand side with Map S ((S 1 ) ⇥n , Map C (X, Y )). On the other hand, since C 0 satisfies the condition (UM) we see that

(S 1 C ) ⇥n ⇥ X ' (S 1
) ⇥n ⌦ X and therefore

Map C ((S 1 C ) ⇥n ⇥ X, Y ) ' Map S ((S 1 ) ⇥n , Map C (X, Y )). Let now X, Y 2 LMod y(S 1 C ) (C 0 ). Using [Lur12c,
F X ([n], 0) F X ([n], 0) (S 1 ) ⇥n y(S 1 C ) ⇥n , e F Y ([n], 0) F Y ([n], 0) (S 1
) ⇥n y(S 1 C ) ⇥n are pullback squares. As already remarked in the proof of Theorem 7.2.2.4, we can describe morphisms in LMod S 1 (C 0 ) as fiber products

Map S 1 (X, Y ) ' Map Fun( op⇥ 1 ,C) (F X , F Y ) ⇥ Map Fun( op,C) (F X | op⇥{1} ,F Y | op⇥{1} ) {id S 1 }
Using the end formula to compute natural transformations, we can describe Map Fun( op⇥

1 ,C) (F X , F Y ) as Z ([n],i)2 op⇥ 1 Map C (F X ([n], i), F Y ([n], i)) ' Z i2 1 Z [n]2 op Map C (F X ([n], i), F Y ([n], i)).
Bringing the fiber product inside the end, we can rewrite

Map S 1 (X, Y ) ' Z i2 1 Z [n]2 op Map C (F X ([n], i), F Y ([n], i)) ⇥ Map C (F X ([n],i),F Y ([n],1)) {p n }
where p n is the identity when i = 1 and the natural projection

p n : F X ([n], 0) ! F X ([n], 1) ' F Y ([n], 1)
when i = 0. A similar description holds for Map y(S 1 C ) (⇢ ⇤ (X), ⇢ ⇤ (Y )). We now observe that the fiber product

Map C (F X ([n], i), F Y ([n], i)) ⇥ Map C (F X ([n],i),F Y ([n],1)) {p n }
is contractible when i = 1 and coincides with Map C ((S 1 ) ⇥n ⇥ X, Y ) when i = 0. Similarly, the fiber product

Map C ( e F X ([n], i), e F Y ([n, i]) ⇥ Map C ( e F X ([n],i), e F Y ([n],1) {e p n }
is contractible when i = 1 and coincides with Map C (y(S 1 C ) ⇥n ⇥ X, Y ) when i = 0. Since we saw that the morphism (7.2.2.6) is an equivalence, we finally conclude that ⇢ ⇤ is fully faithful.

As for essential surjectivity, we observe that giving an object in LMod S 1 (C 0 ) is equivalent to provide a morphism of E 1 -monoids in PSh(C)

S 1 ! Map PSh(C) (y(X), y(X))
, where X 2 C 0 . Using the monoidal adjunction ⇡ p a p ⇡, this is equivalent to give a morphism of E 1 -monoids

S 1 ! Map C (X, X) in S.
Finally, the condition (UM) again implies that such a datum is equivalent to the datum of an action of S 1 C on X. We therefore obtain an object in LMod S 1 C (C 0 ) that induces the S 1 -object in C 0 we started with. Therefore the functor LMod S 1 (C 0 ) ! LMod y(S 1 ) (C 0 ) is essentially surjective. Thus, the proof is complete.

Algebraic HKR theorem

We now put in fruition the technology developed so far to obtain a proof of the HKR theorem in the algebraic setting. We put in motion the general strategy outlined in the introduction.

We denote by CAlg k the 1-category of connective cdgas over k and we let CAlg nc k denote the 1-category of nonconnective ones. In Section 7.2.1 we introduced and studied at length the 1-categories "-CAlg k and "-CAlg nc k . In particular, in Theorem 7.2.1.7 we provided a comonadic description of this category over CAlg nc k , and in Theorem 7.2.1.6 we identified the left adjoint to the forgetful functor with the de Rham algebra functor. The comonad of "-CAlg nc k over CAlg nc k is given by tensor product with the bialgebra k[⌘]. Recall that the algebra structure on k[⌘] is, by definition, the one coming from the split square-zero extension. Let us denote this bialgebra by A " .

On the other hand, in Section 7.2.2 we studied S 1 -CAlg k and S 1 -CAlg nc k . Set C := (CAlg nc k )op. Since the tensor product of nonconnective cdgas commutes with finite limits, we see that the condition (M) (and in fact the stronger condition (UM)) is satisfied. In particular, Theorem 7.2.2.7 guarantees that

S 1 C ' k ⇥ k⇥k k. Let us denote this bialgebra by A S 1 .
Instead of comparing "-CAlg nc k and S 1 -CAlg nc k directly we will compare the two 1-categories

A " -CoMod(CAlg nc k ) and A S 1 -CoMod(CAlg nc k ).
This is enough for our purposes: in fact, Theorem 7.2.1.7 provides us with an equivalence

"-CAlg nc k ' LMod A" (CAlg nc k ).
On the other hand, since C = (CAlg nc k )op satisfies the condition (UM), we see that Theorem 7.2.2.10 provides an equivalence

S 1 -CAlg nc k ' LMod A S 1 (CAlg nc k ).
Warning 7.2.3.1. In the analytic setting the latter equivalence will not be satisfied. However the weaker equivalence

S 1 -CAlg nc k ⇥ CAlg nc k CAlg k ' LMod A S 1 (CAlg nc k ) ⇥ CAlg nc k CAlg k will still hold.
As outlined in the introduction, we are then reduced to compare the bialgebras A " and A S 1 . For this, we introduce a middle comparison term. From a purely algebraic point of view, we simply take Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). The universal property of the symmetric algebra provides us with a canonical zig-zag in CAlg nc k Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF])

A " A S 1 . (7.2.3.1)
Since we are in characteristic 0, we see that the chain complex underlying Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) coincides with k k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. 2Therefore the two morphisms in the above zig-zag are equivalences. We are left to check that these two morphisms can be promoted to morphisms of bialgebras. This is not entirely tautological, and to prove it we need to resort to a more geometrical description of Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). Indeed, let dSt k denote the 1-category of derived stacks over k. This 1-category comes equipped with a global section functor : dSt k ! CAlg nc k , that can be obtained as left Kan extension of the global section functor on affine derived schemes. Then we have a canonical equivalence

Sym k (k[ 1]) ' (B(G a )).
The functor admits a left adjoint, denoted Spec, which can be described as a restricted Yoneda embedding. This functor is not fully faithful, but it becomes so when restricted to the full subcategory of CAlg nc k spanned by coconnective algebras [Toë06a]. In particular, this allows to identify the space of coalgebra structures on Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) (compatible with the given algebra structure) with the space of E 1 -group structures on B(G a ). We have: Proposition 7.2.3.2. The space of E 1 -group structures on B(G a ) is contractible.

Proof. The space X of E 1 -group structures on B(G a ) is defined as the pullback

X Mon gp E1 (dSt k ) {⇤} dSt k . forget B(Ga) (7.2.3.2)
Notice that ⇡ 0 (B(G a )) ' Spec(k), which is the final object in the 1-topos dSt k . In other words, B(G a ) 2 dSt 1 k . Notice furthermore that the inclusion dSt 1 k ,! dSt k commutes with products. It follows that we can split the square (7.2.3.2) into the following ladder of pullbacks:

X Mon gp E1 (dSt 1 k ) Mon gp E1 (dSt k ) {⇤} dSt 1 k dSt k . B(Ga)
It is therefore enough to compute the fiber product on the left. Consider the commutative rectangle

X Mon gp E1 (dSt 1 k ) Mon gp E1 (Mon gp E1 (dSt k )) {⇤} dSt 1 k Mon gp E1 (dSt k ) ⌦ B(Ga) ⌦
May's delooping theorem implies that the horizontal morphism in the square on the right are equivalences. In particular, the square in question is a pullback. As a consequence, it is enough to compute the outer pullback.

Observe now that ⌦(B(G a )) ' G a and that this is a discrete object in dSt k . Furthermore, the induced E 1group structure on G a coincides with the additive one. We now observe that, since G a is discrete and since the E 1 -structure is fixed, being E 1 is now a property rather than a structure. In other words, we see that X is either empty or contractible. As the additive group structure on G a is commutative, we see that it is indeed the latter case.

This proposition implies therefore that the space of bialgebra structure on Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) extending the given algebra structure is contractible. In particular, both morphisms Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) ! A " and Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) ! A S 1 can be promoted to equivalences of bialgebras. Thus, we obtain an equivalence A " ' A S 1 as bialgebras. This implies immediately the structure HKR: Theorem 7.2.3.3. Let k be a field of characteristic zero. There is a natural equivalence :

S 1 -CAlg k ' "-CAlg k making the diagram S 1 -CAlg k "-CAlg k CAlg k ⇠ v S 1 v" commutative.
Proof. Since k has characteristic zero, both maps in the zig-zag (7.

(k[ 1]) ! A S 1 and Sym k (k[ 1]
) ! A " can be promoted to equivalences of bialgebras. This provides us with an equivalence A S 1 ' A " of nonconnective bialgebras. Therefore, we obtain an equivalence

A S 1 -CoMod(CAlg nc k ) ' A " -CoMod(CAlg nc k )
compatible with the forgetful functors to CAlg nc k . Notice that A " -CoMod(CAlg nc k ) coincides by definition with the 1-category we previously denoted ⌘-CAlg nc k . We can therefore invoke Theorem 7.2.1.7 to deduce that

"-CAlg k ' A " -CoMod(CAlg nc k ).
Furthermore, this equivalence is compatible with the forgetful functors to CAlg nc k . On the other hand, Theorem 7.2.2.10 provides us with an equivalence

A S 1 -CoMod(CAlg nc k ) ' S 1 -CAlg nc k ,
also compatible with the forgetful functors to CAlg nc k . Putting all the information together, we obtain the following commutative diagram:

"-CAlg nc k A " -CoMod(CAlg nc k ) A S 1 -CoMod(CAlg nc k ) S 1 -CAlg nc k CAlg nc k . ⇠ v" ⇠ ⇠ v S 1
We now remark that there are canonical equivalences

"-CAlg k ' "-CAlg nc k ⇥ CAlg nc k CAlg k and S 1 -CAlg k ' S 1 -CAlg nc k ⇥ CAlg nc k CAlg k .
The conclusion follows.

Nonconnective contexts and structures

As we saw in the algebraic case, non-connective algebras play a fundamental role in the proof of the HKR theorem. However, one limitation of derived analytic geometry (as introduced in [Lur11b,PY16b]) is that it only allows to work within the connective framework. Indeed, if X = (X, O X ) is a derived analytic space, then the underlying algebra O alg X is always a simplicial commutative ring. The goal of this section is to explain how this problemma can be solved, by introducing a suitable notion of non-connective analytic ring.

Definitions

We work in the general context of pregeometries. This will allow us to obtain several versions of HKR theorems. We refer the reader to [Lur11d, Definition 3.1.1] for the notion of pregeometry. (ii) a choice of mophisms P in C.

Furthermore, we impose the following conditions:

(i) the ⌧ -topology on Top R (T) restricts to a Grothendieck topology on C, which we still denote ⌧ ;

(ii) for every X 2 T the morphism X ! ⇤ is in P;

(iii) C has finite limits;

(iv) the inclusion T ⇢ C commutes with products; ). Then we take C := dAn k , the 1-category of derived k-affinoid spaces. As in the previous example, this is the full subcategory of dAn k spanned by those derived k-analytic spaces X whose truncation t 0 (X) is k-affinoid (cf. [PY16b, Definition 7.3]).

Remark 7.3.1.5. In the case of a generic pregeometry (T, adm, ⌧) there is always a canonical choice for the category C. Indeed, if G denotes a geometric envelope for T in the sense of [Lur11d, Definition 3.4.1], then one can take C to be the 1-category of G-schemes. However, at this level of generality there is no good choice of the collection of morphisms P. One could take the collection of étale morphisms, but this choice would lead to a rather degenerate situation in what follows.

For the following definition, we recall that whenever (C, ⌧, P) is a geometric context, one can define an associated 1-category of geometric stacks. We denote this 1-category by Geom(C, ⌧, P) and we refer to [PY16d, §2.2] for the definition. We also recall that if E is an 1-category with finite limits then Sp(E) ' Sp(E ⇤ ).

In particular, any spectrum object E 2 Sp(E) receives a canonical map ⇤ ! ⌦ 1 n (E), where ⇤ denotes the final object of E.

let G E be the full subcategory of G spanned by the family of objects {(E n , n)} n2N . Since ⌦(E n ) ' E n 1 , we see that the inclusion G E ⇢ G preserves Cartesian edges. Furthermore, the composition

f E : G E G C f is fully faithful.
Using [Lur09c,3.3.3.2] we can identify Sp(Geom(C, ⌧, P)) with 1-category of Cartesian sections of q : C ! Nop. Let s E : Nop ! C be the section determined by E. Notice that s E (n) ' E n . In particular, s E factors through the essential image of f E . Since f E is fully faithful, we can therefore a Cartesian section s 0 E : Nop ! G E such that f E s 0 E ' s E . Composing with the inclusion G E ⇢ G, we can review s 0 E as a Cartesian section of G. In other words, we can associate to E a spectrum object in G nc . In what follows, we denote this spectrum object again by E. This abuse of notation is justified by the full faithfulness of f E .

Let us now fix an 1-topos X. The universal property of G nc produces an equivalence of 1-categories Str T nc (X) ' Fun lex (G nc , X).

Since G nc has finite limits, we can identify Sp(G nc ) with the 1-category Exc ⇤ (S fin ⇤ , G nc ) spanned by strongly excisive functors. In particular, we obtain an evaluation map Exc ⇤ (S fin ⇤ , G nc ) ⇥ Fun lex (G nc , X) ! Exc ⇤ (S fin ⇤ , X).

Evaluation at E 2 Sp(G nc ) provides us with a functor U : Str T nc (X) ! Sp(X). )).

We are therefore authorized to think to the sequence {O(E n )} n 0 as an ⌦-spectrum in X. The above construction, is a formalization of this rough idea. The reader should however observe that the spaces ⌦ 1 n (U (O)) do not have a ring structure. In particular, one cannot interpret U (O) as a spectrum object in the category of simplicial commutative rings. This is due to the fact that there is no multiplication map (ii) a T-structure on ⌦ 1 (A);

(iii) an additional structure on A exhibiting A as an algebra over the ring of cohomology operations associated to E.

In both Theorem 7.3.3.4 and Theorem 7.3.3.5 we used in an essential way the fact that the pregeometry T is discrete. Although this case is sufficient to deal with the algebraic setting and the C-analytic setting, it is unfortunately too restrictive to deal with the rigid analytic setting, at least at the current state of development of derived rigid analytic geometry. The following theorem is a variation of Theorem 7.3.3.5 that removes the discreteness assumption on T at the cost of working only with local T-structures. The proof is conceptually similar to the one of Theorem 7.3.3.4, but we replace the equivalence Str T (X) ' P ⌃ (T) with the existence of sifted resolution by elemmaentary structures, see [Lur11b,Proposition 2.11].

Corollary 7.3.3.6. Let X be an 1-topos with enough points. The connective cover functor ⌧ 0 : Str loc T nc (X) ! Str loc T (X) admits a fully faithful left adjoint.

Proof. Consider the functor

Str loc

T (X) Fun(T, X) Fun(T nc , X).

Lanj

It is enough to prove that this functor factors through Str loc T nc (X). This will provide a functor i : Str loc T (X) ! Str loc T nc (X) and prove that it is fully faithful. Let therefore O 2 Str loc T (X). Since Lan j is fully faithful, we already know that Lan j (O)| T is a local T-structure. It is therefore enough to prove that Lan j (O) commutes with finite products and that for every n 1 the canonical morphism

Lan j (O)(E n ) ! ⌦Lan j (O(E n+1 ))
is an equivalence. Using Theorem 7.3.3.1, we rather prove that for every n 1 the morphism

Lan j (O)(⇤) ! Lan j (O)(E n )
is an effective epimorphism. Using [Lur11b, Proposition 2.11] we choose a Cartesian fibration q : D ! X and a diagram Q : D . ! Str loc T (X) such that: (i) the fibers of q are essentially small sifted 1-categories;

(ii) Q is a colimit diagram relative to X; (iv) the image of the cone point via Q is equivalent to O.

Recall from [Lur11d, Proposition 3.3.1] that the inclusion Str loc T (X) ! Fun(T, X) commutes with sifted colimits. Therefore, reasoning as in Theorem 7.3.3.3, we see that it is enough to prove that if O is an elemmaentary Tstructure, then n Lanj (O) is an equivalence for every n 1. Let therefore Y 2 T and let f : (X, O X ) ! Spec T (Y ) be a fixed morphism. We have to prove that

B Lan j (f 1 (O Y ))(E n ) ! Lan j (f 1 (O Y ))(E n+1 )
is an equivalence. This can be checked on stalks. After passing at the stalk at a geometric point p ⇤ : S X : p 1 , we are reduced to the situation where X = S. In this case, f 1 O Y can be written as a sifted colimit of T-structures of the form A Y↵ . As the functor Lan j commutes with colimits, the conclusion follows from Theorem 7.3.3.2. We will now prove that cdga k and Str T disc (k) nc (S) are equivalent using the Barr-Beck-Lurie criterion for equivalences [Lur12c,4.7.3.16]. We start by constructing a functor between these two categories. Given an unbounded cdga A 2 cdga k , we define a functor Proof. Recall from Theorem 7.3.2.5 that U commutes with limits and filtered colimits. As both Str T disc (k) nc (S) and Mod k are presentable, it follows that U admits a left adjoint, that we denote F . In virtue of [Lur12c,4.7.3.16] it is enough to prove that for every M 2 Mod k the canonical map

(Sym k (M )) ! F (M )
is an equivalence. Notice that the commutativity of (7.3.4.1) coupled with the fact that both U and V are conservative and commute with sifted colimits implies that commutes with sifted colimits as well. In particular, it is enough to prove the statement when M = k[ n] for n 0. In this case, we observe that Sym

Nonconnective cotangent complex

In this section we prove that the adjunction (i, ⌧ 0 ) of Section 7.3.3 induces an equivalence after stabilization. In particular, this allows to introduce nonconnective split square-zero extensions. Fix a pregeometry (T, adm, ⌧) and a T-nonconnective context (C, P, E). Fix also an 1-topos X with enough points. Then the connective cover functor ⌧ 0 : Str T nc (X) ! Str T (X) commutes with limits and filtered colimits, and therefore it has a left adjoint, denoted i. Notice that Corollaries 7.3.3.6 and 7.3.3.7 imply that the functor i is fully faithful when restricted to Str loc T (X). Fix a local T-structure A 2 Str loc T (X). Then ⌧ 0 (i(A)) ' A and we therefore obtain a well defined 1-functor

⌧ 0 : Str T nc (X) /i(A) ! Str T (X) /A .
This functor still commutes with limits and filtered colimits, and therefore it admits a left adjoint, that we still denote i. Now, recall from [Por15b, Corollary 9.4] that Str loc T (X) /A is a presentable 1-category and that the functor Str loc T (X) /A ! Str T (X) /A is fully faithful and admits a left adjoint. Furthermore, after passing to the stabilization we obtain an equivalence Sp(Str loc T (X) /A ) ' Sp(Str T (X) /A ). (7.3.5.1)

By definition, we obtain forgetful functors ⇢ i : Str T nc 01 (X) ! Str T nc i (X), for i = 0, 1. These functors commute with limits and filtered colimits and therefore the adjoint functor theorem guarantees the existence of left adjoints i : Str T nc i (X) ! Str T nc 01 (X).

As a consequence, we obtain two functors As in the case of a single spectrum, the restriction to T inside T nc 01 allows to associate to each T nc 01 -structure a T-structure. We denote this functor once again by ⌧ 0 : Str T nc 01 (X) ! Str T nc (X).

This functor admits a left adjoint, given by left Kan extension along T ,! T nc 01 . This follows from the same argument given in ??. Proof. The commutativity of the diagram follows just from unraveling the definitions. For the left adjointability, we remark that for X 2 T nc i one has an equivalence of comma categories

T B ⇥ T nc 01 {X} ' T B ⇥ T nc i {X}
induced by full faithfulness of both functors in the composition T ,! T nc i ,! T nc 01 . The conclusion now follows because the adjoint to ⌧ 0 is computed in both cases by a plain left Kan extension.

It is a more subtle question to understand whether the same properties hold for the functors i . i ,! T nc 01 is typically asymmetric. This means that it will often be satisfied for only one of the two inclusions T nc 0 ,! T nc 01 , T nc 1 ,! T nc 01 .

Proof. First we remark that, since colimits in PSh(C 0 ) are computed objectwise and since n-truncated objects are stable under filtered colimits, the presheaf F 1 is n-truncated. Let now X 2 C 0 be an object and let U • be a ⌧ -hypercover of X. Since the topology ⌧ is quasi-compact, we can suppose that for every [m] 2 , U m is disjoint union of finitely many objects in C 0 . We have to prove that the canonical map

F 1 (X) ! lim F 1 (U •
) is an equivalence. Since F (X) and F 1 (U m ) are n-truncated for every [m] 2 and since S m is closed under limits in S, we see that we can compute the above limit in S n . Since S n is an n-category and n+2 ,! is n-cofinal, we see that there is a canonical equivalence

lim F 1 (U • ) ' lim n+2 F 1 (U • ).
It is therefore enough to prove that the canonical map from F (X) to the right hand side is an equivalence. Notice that, since each U m is a finite disjoint union of objects in C 0 , the limit on the right is a finite. Since filtered colimits commute with finite limits, we have a canonical equivalence

lim n+2 F 1 (U • ) ' colim ↵2I lim n+2 F ↵ (U • ).
Since each F ↵ is n-truncated, we can use once more the n-cofinality of the inclusion n+2 ,! to deduce that the canonical map

F ↵ (X) ! lim n+2 F ↵ (U • )
is an equivalence. The conclusion follows.

This is provided by our next result: ).

We are therefore left to check that Lan(O) commutes with finite products. This follows from an analysis case by case as in the proof of ??.

It is therefore enough to prove the claim. We first observe that since a is a morphism of E 1 -groups, the 1-categorical version of May's theorem implies that it induces an endomorphism is an equivalence. When X belongs to T, this is a direct consequence of Theorem 7.3.6.9. In the general case, we can find an hypercover U • of X such that each U • is a disjoint union of finitely many objects in T. In particular, we have equivalences

Map

Geom(C,⌧,P) (X, E n i ) ' lim Map Geom(C,⌧,P) (U • , E n i ),

for i = 0, 1. Since E n i is n-truncated in virtue of Theorem 7.3.6.8, we see that the above is a limit in S n . As S n is an n-category and n+2 ,! is n-cofinal, we conclude that we also have an equivalence

Map

Geom(C,⌧,P) (X, E n i ) ' lim n+2 Map Geom(C,⌧,P) (U • , E n i ).

The conclusion now follows from the fact that each U m is a finite disjoint union of finitely many objects in T, from Theorem 7.3.6.9 and from the fact that filtered colimits commute with finite limits. We now consider the comma category (T nc 0 ) /E n 1 . The above argument implies that the diagram N ! (T nc 0 ) /E n 1 corresponding to the iteration of the morphism a n : E n 0 ! E n 0 is cofinal. Therefore, the claim follows from the explicit formula for left Kan extensions. Str T nc i (X) /A @(⇢i)

⌦ 1 ⌦ 1 ⇢i
commutes and it is left adjointable. In particular, the functor 01 : Str T nc 0 (X) ! Str T nc 1 (X) commutes with the formation of split square-zero extensions.

Proof. The commutativity of the diagram simply follows from the fact that the functor ⇢ i commutes with limits. For the left adjointability, we observe that the proof of Theorem 7.3.6.3 implies that the functor Lan commutes with finite limits, because it can be computed by filtered colimits. The conclusion follows.

Theorem 7.3.8.3. Let (T 0 , adm 0 , ⌧ 0 ) and (T 1 , adm 1 , ⌧ 1 ) be two pregeometries. For i = 0, 1, let (C i , P i , E i ) be a pre-T i -nonconnective context. Let ' : (C 0 , P 0 , E 0 ) ! (C 1 , P 1 , E 1 ) be a Morita equivalence. Suppose furthermore that the restriction ' : T 0 ! T 1 satisfies the following properties:

(i) ' : T 0 ! T 1 is fully faithful;

(ii) a morphism f : U ! X in T 0 is admissible if and only if '(f ) : '(U ) ! '(X) is admissible;

(iii) a collection of admissible morphisms {f ↵ : U ↵ ! X} generates a covering sieve on X in T 0 if and only if the collection {'(f ↵ ) : '(U ↵ ) ! '(X)} generates a covering sieve on '(X) in T 1 ;

(iv) for every X 2 T 1 there exists a collection of objects {U ↵ } in T 0 and a collection of morphisms {f ↵ : '(U ↵ ) ! X} generating a covering sieve.

If in addition (C 0 , P 0 , E 0 ) is a T 0 -nonconnective context, then for any 1-topos X the restriction functor T nc 1 (X) and that such factorization is an equivalence. Remark 7.3.8.4. It is possible to push the theory of nonconnective pregeometries much farther than what we did so far. For example, it would be possible to develop a theory of nonconnective schemes, which would provide an analogue of the theory of spectral affine schemes introduced in [Lur11f]. Having a notion of nonconnective scheme for a suitably general pregeometry might be useful: for example, the paper [AHR10] suggests the existence of a complex analytic version of tmf. In order to make rigorous the considerations done in loc. cit. it is necessary to consider nonconnective derived analytic spaces. This can easily be achieved by the formalism of nonconnective pregeometries. Another possible application of nonconnective structures would be to develop analogues of the results in [Toë06a] in the complex analytic and in the non-archimedean analytic setting.

The analytic case

From this point on we specialize to the analytic setting. Our first task is to introduce the categories of mixed and S 1 -equivariant analytic algebras.

In this section we let k denote either the field of complex numbers or a non-archimedean field equipped with a non-trivial valuation and of characteristic zero. In the latter case, we let | | denote the associated absolute value. Following the use of [PY16d,PY17b], we write C-analytic to mean complex analytic and k-analytic to mean non-archimedean analytic over the non-archimedean field k. When statements apply to both settings, we simply write analytic. Notice that the inclusion (dAfd k , ⌧ ét , P sm ) ,! (dAn k , ⌧ ét , P sm ) induces an equivalence of the 1-categories of geometric stacks thanks to [PY16d,Corollary 2.26].

Let now BG a denote the analytic affine line equipped with its additive group structure. Since BG a is commutative, May's delooping theorem for 1-topoi [Lur12c, 5.2.6.15] provides us with a spectrum object E 2 Sp(Geom(dAn k , ⌧ ét , P sm ))

such that E n ' B n (BG a ). In this way, (dAn k , ⌧ ét , P sm , E) becomes a pre-T an (k)-nonconnective context, while (dAfd k , ⌧ ét , P sm , E) becomes a T afd an (k)-nonconnective context. The latter assertion is a consequence of the derived version of Tate's acyclicity theorem (see [PY18a,Theorem 3.4]). In particular, (dAfd k , ⌧ ét , P sm , E) defines a nonconnective pregeometry, that accordingly to our convention we should denote T afd an (k) nc . However, when no confusion is possible we simplify the notation and denote it instead by T nc an (k).

complex are (co)homologous if and only if the boundary element corresponds to a section uniformly bounded in norm by r on the intersections and thus on all X. This implies that we have a monomorphism of graded modules

⇡ ⇤ ( (t 0 X, ⇡ 0 O X (r))) ! ⇡ ⇤ ( (t 0 X, ⇡ 0 O X ))
and therefore, we conclude that the mapping space in (7.4.1.1) is discrete, as desired.

Therefore, the latter defines a pregeometry, that we denote by T nc an (k; r).

Nonconnective analytification

In [Por15b,PY17b] the authors studied and exploited the derived analytification functor. This is a functor that associates to every derived scheme locally almost of finite presentation over k a derived analytic space, which is characterized by a certain universal property. At the level of algebras, the analytification functor can simply be understood as the left adjoint to the underlying algebra functor. In our approach to the analytic HKR theorem, the analytification functor plays a major role, for instance in the definition of mixed analytic algebras.

The classical analytification functor induces a transformation of pregeometries ( ) an : T ét (k) ! T an (k).

The analysis carried over in [Por15b,[START_REF] Bommel | The Grothendieck monodromy theorem[END_REF] and in [PY17b,[START_REF] Bhatt | The pro-étale topology for schemes[END_REF] shows that this functor extends to a transformation of geometric contexts ( ) an : (dA↵ afp k , ⌧ ét , P sm ) ! (dAn k , ⌧ ét , P sm ). In particular, [PY16d, Proposition 2.25] provides us with a well defined functor ( ) an : Geom(dA↵ afp k , ⌧ ét , P sm ) ! Geom(dAn k , ⌧ ét , P sm ).

Notice that this functor commutes with colimits by construction, and therefore that it brings B n (G a ) to B n (BG a ). As a consequence, we obtain a strong transformation of nonconnective pre-contexts ( ) an : T nc ét (k) ! T an (k) nc .

Remark 7.4.2.1. In the C-analytic setting, T an (k) nc is a nonconnective context. On the other hand, in the rigid analytic setting, T an (k) nc is only a nonconnective pre-context. However, the inclusion T afd an (k) nc ,! T an (k) nc satisfies the assumptions of ??, therefore providing for every 1-topos X an equivalence Str loc T afd an (k) nc (X) ' Str loc Tan(k) nc (X).

We denote this 1-category simply by Str loc T nc an (k) (X). The analytification functor ( ) an : T nc ét (k) ! T an (k) nc fits into the following commutative square

T ét (k) T an (k)
T nc ét (k) T an (k) nc .

( ) an

( ) an

Therefore, for every 1-topos X, we obtain a commutative diagram A key feature of the underlying algebra functor on local structures is the fact that it is conservative. Let us record this property explicitly for later use: Proof. In the connective case, this has already been proven in [Lur11b, Proposition 11.9] (in the C-analytic setting) and in [PY16b, Lemma 3.13] (in the rigid analytic setting). In the nonconnective situation, the statement follows at once from the connective one and from Theorem 7.3.3.8.

For the rest of this section, we restrict ourselves to the 1-topos of spaces S. In this case, the Yoneda lemma allows to produce several important examples of nonconnective analytic structures. Indeed, for every derived analytic stack X 2 dAnSt k , we can define a functor an k (X) : T nc an (k) ! S by an k (X)(U ) := Map dAnSt k (X, U ). This is a T nc an (k)-structure, but in general it is not local. The following proposition isolates a special class of derived analytic stack X for which an k (X) is local.

Proposition 7.4.2.5. Suppose that X is underived and ⇡ 0 (X) ' Sp(k). Then an k (X) is a local T nc an (k)-structure. Proof. We only need to check that ⌧ 0 ( an k (X))) is a local T an (k)-structure. Let U 2 T an (k). Since X is underived, we have

Map

dAnSt k (X, U ) ' Map AnSt k (X, U ). Since U is discrete, we have

Map

AnSt k (X, U ) ' Map AnSt k (⇡ 0 (X), U) ' Map AnSt k (Sp(k), U).

This functor clearly sends ⌧ ét -coverings to effective epimorphism.

In particular, the objects an k (E n ), an k (E(r) n ) 2 Str T nc an (k) (S) (r 2 |k| ⇥ ) are local structures. We denote them by S an (n) and S an k (n, r) respectively and we refer to them as the free nonconnective analytic algebra of rank 1, degree n (and radius r). Notation 7.4.2.6. We denote by AnRing nc k the 1-category Str loc T nc an (k) (S) /k . We refer to it as the 1-category of local nonconnective analytic rings. Notice that it is a presentable 1-category. In particular, it admits pushouts. Given maps A ! A 0 and A ! A 00 we denote their pushout by A 0 b ⌦ A A 00 .

Notice that in the algebraic case, the identification of Str Proof. This simply follows from the fact that (B n (G a )) an ' B n (BG a ).

It is less trivial to identify (S an k (n, r)) alg . We start by observing that for every r 2 |k ⇥ | there is a natural morphism B n (D 1 (r)) ,! B n (BG a ), that induces a (local) morphism of nonconnective analytic rings S an k (n) ! S an k (n, r).

On the other hand, the unit of the adjunction ( ) an a ( ) alg provides us with a map ⌘ n,r : S k (n) ! (S an k (n)) alg ! (S an k (n, r)) alg

The following result is the key to the analytic HKR: Proof. Since the underlying algebra functor ( ) alg : AnRing nc k ! cdga k is conservative, the second statement follows at once from the first one.

Since the forgetful functor cdga k ! Mod k is conservative, it is enough to check that the image of ⌘ n,r in Mod k is an equivalence. It follows from the definitions that this is the same morphism obtained by applying the underlying spectrum functor to ⌘ n,r . Unraveling the definitions, we see that the object U (S an k (n, r)) 2 Mod k is computed as the totalization of the following cosimplicial object:

k khr 1 T i khr 1 T 1 , r 1 T 2 i • • • ,
which in degree n has khr 1 T 1 , . . . , r 1 T n i, and the ith morphism

T j+1 if j < i T j + T j+1 if j = i T j if j > i.
When r = 1, we denote by khr 1 T 1 , . . . , r 1 T n i the algebra of analytic functions on A n k . Let us first deal with the complex analytic case. In this case r = ⇢ = 1. After applying the cosimplicial Dold-Kan, we can identify the above cosimplicial object with the cochain complex computing holomorphic cohomology of BG a with coefficients in its trivial representation of rank 1. The Van Est theorem (originally formulated for continuous cochains and extended to holomorphic cochains by Hochschild and Mostow) implies that we can identify the above complex with the complex computing the cohomology of the Lie algebra g a of BG a with coefficients in its trivial representation of rank 1. Inspection reveals that the latter is quasi-isomorphic to k k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. The conclusion now follows from ??.

In the non-archimedean analytic case the same thing happens, with the difference that the cochain complex is no longer acyclic in degrees  2 unless r  ⇢. This has originally been proven by Lazard. See [?] for a modern treatment.

We also need a relative version of Van Est theorem. The map ⌘ n,r induces a well defined map S A (n) ! (S an A (n, r)) alg , that we still denote ⌘ n,r .

Corollary 7.4.2.9. Let ⇢ 2 |k| ⇥ [ {1} denote the converging radius of the exponential. Then if r  ⇢ and A 2 AnRing k , the map ⌘ n,r : S A (n) ! (S an A (n, r)) alg is an equivalence.

Proof. We start by observing that the formation of the map ⌘ n,r : S A (n) ! (S an A (n, r)) alg commutes with filtered colimits in A. Using ??, we reduce ourselves to the case where A ' x 1 O X , for X = (X, O X ) 2 dAn k and x ⇤ : S ⌧ X : x 1 a geometric point.

Notice that x 1 O X b ⌦ k S an k (n, r) ' x 1 (O X b ⌦S an X (n, r)). we can reduce ourselves to the case where A is the germ of analytic functions at a point in the interior of the disk D m k . As the formation of the map ⌘ n,r commutes with filtered colimits, we see that it is enough to prove that it is an equivalence replacing A with the analytic ring of overconvergent analytic functions on D m k . In this case the Yoneda lemma allows to identify the completed tensor product A b ⌦ k S an A (n, r) with the analytic ring associated to the stack (D m k ) † ⇥ B n (D 1 k (r)), and hence with the inverse limit lim

✓ A A b ⌦ k S an k (n 1, r) A b ⌦ k (S an k (n 1, r)) b ⌦ 2 • • • ◆ .
As the underlying algebra functor commutes with inverse limits, we can reason by induction on n, and we are immediately reduced to the case n = 1. In this case T l i of functional analysis. At this point, the conclusion follows because A is Fréchet nuclear: see [Dem,Example 5.12] in the C-analytic case and [BBB15] in the kanalytic case. In particular, the completed tensor product with A is acyclic. Thus, the conclusion follows directly from Theorem 7.4.2.8.

A nonconnective base change

Before starting to discuss S 1 and mixed analytic algebras, we need a couple of preliminaries on the general features of nonconnective analytic structures. We start with a discussion of a very particular base change property.

Notice that we have a functor There is a third alternative way to describe the derived analytic stack (S 1 ) an . We can in fact identify it with the delooping of the constant stack (Z) an associated to Z, seen as a discrete topological space. As Z is initial among discrete groups, we see that there is a canonical morphism of group stacks (Z) an ! D 1 (r).

In particular, applying the delooping functor provides us with a morphism (S 1 ) an ! B(D 1 (r)).

Passing to global sections, we obtain a morphism of local nonconnective analytic rings ⇡ r : S an k (n, r) ! an k (S 1 ). ) ! an k (S 1 ) alg is an equivalence. In order to see this, we only need to prove that for every n 0, one has

Map dAnSt k ((S 1 ) an , B n (BG a )) ' Map dSt k (S 1 , B n (G a )).
As S 1 ' ⇤ q ⇤q⇤ ⇤, we are reduced to check that As in ??, we are looking for a reformulation in terms of analytic comodules. In order to do this, we need first to discuss the structure of coalgebra of an k (S 1 ).

Proposition 7.4.5.3. Let X 2 dAnSt k be a derived analytic stack. Consider the cartesian diagram X ⇥ (S 1 ) an (S 1 ) an X Sp(k) q 0 p 0 p q in dAnSt k . Then the induced Beck-Chevalley transformation q ⇤ p ⇤ O (S 1 ) an ! p 0 ⇤ q 0⇤ O (S 1 ) an in an equivalence in Str T nc an (k) (X). Proof. Recall that S 1 ' ⇤ q ⇤q⇤ ⇤. Hence (S 1

) an ' Sp(k) q Sp(k)qSp(k) Sp(k), and therefore

p ⇤ O S 1 ' k ⇥ k⇥k k,
the product being taken in Str T nc an (k) (S). Similarly, since colimits in dAnSt k are universal, we have X ⇥ (S 1 ) an ' X q XqX X and therefore p 0 ⇤ q 0⇤ O S 1 ' p 0 ⇤ O X⇥(S 1 ) an ' O X ⇥ O X ⇥O X O X , the product being taken in Str T nc an (k) (X). We are therefore reduced to prove that the canonical morphism

O X b ⌦ k q 1 (k ⇥ k⇥k k) ! O X ⇥ O X ⇥O X O
X is an equivalence. We now invoke Theorem 7.4.5.2 to obtain an equivalence of T nc an (k)-structures k ⇥ k⇥k k ' S an k (1, r), where r is less than the converging radius of the exponential. Using Theorem 7.4.3.1 we now identify O X b ⌦ k q 1 (k⇥ k⇥k k) with the pushforward along X ⇥ B(D 1 k (r)) ! X of the structure sheaf of X ⇥ B(D 1 k (r)), i.e. S an O X (1, r). Using Theorem 7.4.2.9, we can further identify the underlying algebra of S an O X (1, r) with Sym O alg X (O alg X [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). As the underlying algebra functor commutes with limits, we are therefore reduced to check that the canonical morphism

Sym O alg X (O alg X [ 1]) ! O alg X ⇥ O alg X ⇥O alg X O alg
X is an equivalence. Using [Lur11b, Proposition 2.11], we can further reduce to the case where X 2 T an (k). In this case we can identify the underlying module of both sides with O alg X O alg X [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF], whence the conclusion.

Remark 7.4.5.4. Observe that p 0 ⇤ q 0⇤ O S 1 (and hence q ⇤ p ⇤ O S 1 ) belongs to Str loc T nc an (k) (X). Corollary 7.4.5.5. Let D be the smallest full subcategory of dAnSt k closed under products and containing T an (k) and (S 1

) an . Then the restriction In general, we can write X ' X 0 ⇥ ((S 1

) an ) ⇥n and Y ' Y 0 ⇥ ((S 1 ) an ) ⇥m , where X 0 , Y 0 2 T an (k). In this case, the conclusion follows by induction on n and m, using Theorem 7.4.5.3 to deal with the induction step.

As a consequence of this corollary, we can promote an k (S 1 ) to an object in Comon E1 (Str T nc an (k) (S)). Observe furthermore that an k ((S 1 ) ⇥n ) belongs to Str loc T nc an (k) (S). We can therefore review an k (S 1 ) as an object in Comon E1 (AnRing nc k ). If X is an 1-topos and p ⇤ : X ⌧ S : p 1 is the canonical geometric morphism, the functor T nc an (k) (X)). Proof. This is just a consequence of the comonadic version of Barr-Beck-Lurie's theorem.

Theorem 1 .

 1 Soit X un schéma propre et lisse sur un corps algébriquement clos. Alors le foncteur LocSys `,n (X) 2 Fun Afd op Q `, Grpd est représentable par un champ Q `-analytique géométrique. En plus, LocSys `,n (X) admet un atlas lisse par un espace rigide Q `-analytique LocSys framed `,n (X) ! LocSys `,n (X), où LocSys framed `,n

Remark 1 . 3 . 1 . 2 .

 1312 We can extend LocSys C,n (X) to a moduli stack via the formula LocSys C,n (X) : A↵ C ! S given on objects by the formula A 2 A↵ C 7 ! Map S X(C), BGL n (A) 2 S. (1.3.1.1)

1. 4

 4 Derived k -adic and derived k-analytic geometries 1.4.1 Derived Raynaud localization theorem A fundamental ingredient in both Theorem 1.3.3.1 and ?? 1 consists of the rigidification functor ( ) rig : fSch k ! An k (1.4.1.1)

  Theorem 2.1] implies that the functor ( ) + displayed in (1.4.1.2) admits a right adjoint( ) rig : Top R (T ad (k )) ! Top R (T an (k))which is the natural candidate for a derived rigidification functor. Indeed one has the following results: Proposition 1.4.1.10. The functor ( ) rig : Top R (T ad (k )) ! Top R (T an (k)) sends derived k -adic Deligne-Mumford stacks to derived k-analytic spaces, i.e. more specifically ( ) rig restricts to a well defined functor of 1-categories ( ) rig : dfDM k ! dAn k .

Theorem 1 . 4 . 1 . 13 (

 14113 Derived Raynaud localization theorem). The rigidification functor ( ) rig : dfSch k ! dAn k factors through the localization 1-category dfSch k [S 1

Theorem 1 . 4 . 2 . 3 .

 1423 Let f : X ! Y be a flat morphism of derived k-analytic spaces. Then there exists a flat morphism f : X ! Y of derived k -adic Deligne-Mumford stacks such that there exists an equivalence f ' f rig in the 1-category of morphisms dAn 1 k .

Theorem 1 . 4 . 2 . 4 .

 1424 Let dfSch taft,qcqs k denote the 1-category of qcqs topologically almost of finite presentation derived k -adic schemes. Then the functor Coh + loc : dfSch taft,qcqs k op ! Cat st 1 is a hypercomplete sheaf for the Zariski topology. Theorem 1.4.2.4 is a generalization of [HPV16a, Theorem 7.3] to the case where the base is assumed to be derived. With Theorem 1.4.2.4 at hand one is allowed to prove the following important results: Proposition 1.4.2.5. Let X 2 dAn k be a derived k-analytic space. Let F 2 Coh b (X) be a bounded almost perfect module over X. Then for any formal model X 2 dfDM k of X, there exists G 2 Coh b (X) such that G rig ' F in the 1-category Coh + (X). Moreover, the full subcategory of Coh b (X) ⇥ Coh b

  RLocSys `,n (X) := ev 1 C n (Z) , where C n (Z) ✓ Perf(Z) denotes the full subcategory spanned by rank n objects in Perf(Z).

. 5 . 4 . 2 )!!!

 542 Both (1.5.4.1) and (1.5.4.2) imply the existence of a canonical morphism of the form Map 1, Ad(⇢) ⌦ Map 1, Ad(⇢) can Map 1, Ad(⇢) ⌦ Ad(⇢) mult Map 1, Ad(⇢) tr Map 1, 1).

Remark 2 . 1 .0. 6 .

 216 The object O 2 AnRing k (X) does admit more structure than its algebraic counterpart O alg 2 CAlg k (X). For example, we have an induced morphism of derived rings on X O(B 1 k ) ! O alg which one should interpret as the inclusion of radius-1 convergent holomorphic glocal sections on the sheaf of all global sections O alg on X. Therefore, O alg admits a k-analytic structure which cannot be recovered solely by the algebraic structure on O alg . Definition 2.1.0.7. The 1-category of T an (k)-structured spaces, denoted Top R (T an (k)), is defined as the 1category of those couples (X, O) where X 2 Top R is an 1-topos and O : T an (k) ! X is a local T an (k)-structure on X, see [Lur11c, Definition 3.1.9] for a more rigourous construction of Top R (T an (k)).

R(T

  an (k)) to an ordinary k-analytic space, by means of Theorem 3.2.3.2.

3. 3 . 1

 31 Derived k -adic spaces Definition 3.3.1.1. Denote T ad (k ) the full subcategory of the category of k -formal schemes spanned by those formal affine schemes which are formally étale over some A n k . We consider T ad (k ) as a pregeometry by defining the class of admissible morphisms on T ad (k ) to be the class of étale morphisms. We equip T ad (k ) with the étale topology. Notation 3.3.1.2. Denote by Top R (T ad (k )) the 1-category of T ad (k )-structured 1-topoi. Given X 2 Top R an 1-topos we define fCAlg k (X) := Str loc

Remark 3 . 3 . 1 . 4 .

 3314 Let X 2 Top R be an 1-topos. The underlying algebra functor ( ) alg

Remark 3 . 3 . 2 . 7 .

 3327 Given A 2 CAlg ad k , it follows immediately by the definitions that Spf(A) alg 2 Top R (T disc (k )) agrees with the Spf-construction introduced in [Lur16, §8.1.1] Remark 3.3.2.8. Let n 1 and consider the right adjoint functor

  be a morphism of T ad (k )-structured 1-topoi such that (F ) ' . It follows by[Lur16, Remark 8.1.1.7] that the induce geometric morphism (f 1 , f ⇤ ) : X A ! Y B can be identified with the restriction to closed subtopoi of the geometric morphism of 1-topoi X A ! Y B . Thanks to the proof of [Lur16, Proposition 1.4.2.4] it follows that the latter is uniquely determined up to a contractible space of choices. For this reason (f 1

Definition 3 .

 3 3.2.10. A derived k -adic Deligne-Mumford stack is a couple (X, O) 2 Top R (T ad (k )) such that (X, O alg ) formal derived Deligne-Mumford stack as in [Lur16, Definition 8.1.3.1]. We say that a derived k -adic Deligne-Mumford stack (X, O) is topologically almost of finite presentation if the underlying 1-topos X is coherent (cf. [Lur11e, §3]) and the T ad (k )-structure O 2 fCAlg k (X) is topologically almost of finite presentation. Notation 3.3.2.11. We denote dfDM k (resp., dfDM taft k ) the full subcategory of Top R (T ad (k )) spanned by derived k -adic Deligne-Mumford stacks (resp., topologically almost of finite presentation k -adic Deligne-Mumford stacks).

Remark 3 . 3 . 2 . 13 .

 33213 The functor Spf : CAlg ad k ! Top R (T ad (k )) factors through the fully faithful embedding dfSch ,! Top R (T ad (k )).

Proposition 3 . 3 . 3 . 4 .

 3334 Suppose X has enough geometric points and (O alg

Remark 3 . 3 . 3 . 5 .

 3335 The equivalence of stable 1-categories provided in Theorem 3.3.3.4 allow us to define a tstructure on the 1-category Mod O by means of the functor g alg . Definition 3.3.3.6. Let X = (X, O) 2 Top R (T ad (k )). We define the 1-category of coherent O-modules on X, denoted Coh + (X) as the full subcategory Mod O spanned by those F such that for each integer i the homotopy sheaves ⇡ i (F) are coherent ⇡ 0 (O alg

Definition 3 . 3 . 4 . 1 .

 3341 Let M 2 Mod O we shall refer to O M := ⌦ 1 ad (M ) as the trivial adic square-zero extension of O by the module M . Definition 3.3.4.2. Let X := (X, O) 2 Top R (T ad (k )) and let A 2 fCAlg k (X) /O be a T ad (k ) structure on X. Given M 2 Mod O , we define the space of A-linear adic derivations of O with values in M as Der ad A (O, M) := Map fCAlg k (X) A//O (O, O M ) 2 S. Proposition 3.3.4.3. The functor Der ad A (O, ) : Mod O ! S, is corepresentable by an object L ad O/A 2 Mod O which we refer to as the adic cotangent complex relative to O ! A.

Proposition 3 . 3 . 4 . 4 .

 3344 Let A ! B be a morphism in fCAlg k (X) /O topologically almost of finite presentation. Then L ad B/A is a compact object in the 1-category Mod O . Proof. The proof of [Lur16, Proposition 4.1.2.1] applies. Remark 3.3.4.5. Notice that we have a commutative diagram of 1-categories

Proposition 3 . 3 . 4 . 8 .

 3348 Suppose we are given a pushout diagram A B C D in the 1-category fCAlg k (X) /O . Then the natural morphism

Remark 3 . 3 . 5 . 3 .

 3353 The above definition is equivalent to require that given a couple (X, O) 2 Top R (T), the truncation (X, ⌧ n O), where ⌧ n : X ! X denotes the n-truncation functor on X, is again an object of the 1-category Top R (T), or in other terms, ⌧ n O : T ! X is still a T-structure on X. Notation 3.3.5.4. Let T be a preogeometry compatible with n-truncations. We will denote Top R (T) n ✓ Top R (T) the full subcategory spanned by those couples (X, O) such that the T-structure O : T ! X is n-truncated. Remark 3.3.5.5. The inclusion functor Top R (T) n ✓ Top R (T) admits a right adjoint t n : Top R (T) ! Top R (T) n which is given on objects by the formula (X, O) 2 Top R (T) 7 ! (X, ⌧ n O) 2 Top R (T) n . Lemma 3.3.5.6. The pregeometry T ad (k ) is compatible with n-truncations. Proof. We follow closely [Lur11c, Proposition 4.3.28]. Reasoning as in the proof of the cited reference or as in the proof of Theorem 3.3.1.11 it suffices to prove the following assertion: let U ! V be an admissible morphism in T ad (k ) and O 2 fCAlg k (S) then the commutative square

.5. 1 )

 1 is a pullback square in the 1-topos S. By the definition of T ad (k ), there are t-complete ordinary k -algebras A and B such that U ⇠ = Spf A and V ⇠ = Spf B. Moreover, by construction, B is étale over some ring of the form k hT 1 , . . . T m i. [dJ + , Tag A0R1, Lemma 8.0.10.3] implies that there exists an étale k [T 1 , . . . T m ]-algebra

  3.3.5.4) is a pullback diagram in the 1-category fCAlg k (X) /A . Thanks to Theorem 3.3.2.4, one concludes that the diagram (3.3.5.4) is equivalent to the pullback diagram(3.3.5.3). Therefore, the canonical map B 0 ! B is an equivalence in the 1-category fCAlg k (X) /O , as desired.

of 1 -

 1 categories, where both 1-categories Top R (T an (k)) n and Top R (T ad (k )) n are as in Theorem 3.3.5.4. Proof. It follows immediately from the fact that both preogemetries T an (k) and T ad (k ) are compatible with n-truncations, see [PY16a, Theorem 3.23] and Theorem 3.3.5.6. Corollary 3.4.1.3. Let n 1 be an integer. The diagram It follows by taking right adjoints in the diagram displayed in Theorem 3.4.1.2.

Proposition 3 . 4 . 1 . 6 .

 3416 Let X be a topological almost of finite presentation derived k -derived Deligne-Mumford stack. Then X rig is a derived k-analytic space. Proof. Our proof is inspired on [PY17a, Proposition 3.7]. The question is étale local by [Lur11c, Lemma 2.1.3].

  (i) The 1-category C contains the essential image of T ad (k ) thanks to [Lur11c, Proposition 2.3.18]. (ii) C is closed under pullbacks along closed immersions: Let diagram in the 1-category dfDM k such that X, Y and Z 2 C and that f : Y ! X is a closed immersion. By unramifiedness of the pregeometry T ad (k ), Theorem .2.0.3, the diagram (3.4.1.2) is also a pullback diagram in the 1-category Top R (T ad (k )). As ( ) rig is a right adjoint the diagram diagram in the 1-category Top R (T an (k)). The 1-category dAn is closed under pullbacks along closed immersions thanks to [PY16a, Proposition 6.2]. Theorem 3.4.1.5 then implies that the diagram (3.4.1.3) is a pullback square in the 1-category dAn. Thus W 2 C, as desired. (iii) The 1-category C is closed under finite limits. It suffices to prove that C is closed under finite products and pullbacks. [PY16a, Lemma 6.4] implies that C is closed under finite products. General pullback diagrams can be constructed as pullbacks along along closed immersions as in the proof of [PY16a, Theorem 6.5]. Thanks to Theorem 3.4.1.4, ( ) rig commutes with finite products of ordinary formal schemes and preserves closed immersions by Theorem 3.4.1.5, the assertion follows. (iv) C is closed under retracts: let X 2 C and let Y X Y, j p be a retract diagram in the 1-category dfDM k . Assume further that Y is affine. By assumption, X rig 2 dAn and t 0 (Y) rig 2 dAn thanks to Theorem 3.4.1.4. It suffices to prove that for each i > 0, the homotopy sheaf

Remark 3 . 4 . 2 . 2 .

 3422 Notations as in Theorem 3.4.2.1 and suppose further that X is a derived k-analytic space and X 2 dfDM taft k . Thanks to Theorem 3.4.1.4 the geometric morphism underlying f : X + ! X corresponds to the classical specialization morphism at the level of 1-topoi X ! Z. Notation 3.4.2.3. We will denote the geometric morphism introduced in Theorem 3.4.2.1 (f 1 , f ⇤ ) : X ! Z by sp = (sp 1 , sp ⇤ ). Construction 3.4.2.4. Notations as in Theorem 3.4.2.1. Consider the following square of pregeometries

Proposition 3 . 4 . 3 . 2 .

 3432 [PY17a, Theorem 4.5] There exists a canonical equivalence of 1-categories

  k and let X = (X, O) := X rig denote its rigidification. Then the functor ( ) rig : Mod O0 ! Mod O is t-exact. Proof. The statement follows readily from Theorem .1.1.4 and [HPV16a, Corollary 2.9].

k

  the full subcategory of CAlg ad k spanned by admissible adic derived k -algebras. Definition 3.4.4.2. Let X 2 dfDM k we say that X is a derived admissible k -adic Deligne-Mumford stack (or derived admissible k -adic scheme) if X 2 dfDM taft k (resp., X 2 dfSch taft ) and we can find a covering a i Spf(A i ) ! X such that for each i, A i 2 CAlg adm k . We denote by dfDM adm k (resp., dfSch adm ) the 1-category of derived admissible k -adic Deligne-Mumford stacks (resp. derived admissible k -adic schemes).

  denote the full subcategory of C /D spanned by those objects (C, : F (C) ! D) such that is an equivalence in D. Suppose further that C 0 /D is non-empty and moreover the inclusion C 0 /D ! C /D is cofinal. Then F : C ! D induces an equivalence of 1-categories C[S 1 ] ! D, where S denotes the class of morphisms f 2 C 1 such that F (f ) is an equivalence.

Remark 3 . 4 . 4 . 7 .

 3447 Theorem 3.4.4.6 implies that the localization functor of classical Raynaud theorem is 1categorical, i.e. the usual category An 0 of quasi-paracompact and quasi-separated k-analytic spaces is the 1categorical localization of fSch k . This is not a common phenomenon: if C is a 1-category and S a collection of morphisms in C then the 1-categorical localization C[S 1

Theorem 3 .

 3 4.4.10 (Derived Raynaud Localization Theorem). Let S denote the saturated class of morphism of dfSch adm generated by those generically strong morphisms f : X ! Y such that t 0 (f ) is an admissible blow-up of ordinary k -adic schemes. Then the rigidification functor ( ) rig : dfSch adm ! dAn 0 induces an equivalence of 1-categories dfSch adm [S 1 ] ' dAn 0 . Theorem 3.4.4.10 is an immediate consequence of Theorem 3.4.4.6 together with the following Proposition: Proposition 3.4.4.11. The rigidification functor ( ) rig : dfSch adm ! dAn 0 satisfies the dual assumptions of the statement in Theorem 3.4.4.6.

  is an ordinary k -adic formal scheme. The result is now a direct consequence of [Bos05, Theorem 3, page 204].

  whose transition morphisms correspond to (suitable) base change functors. Let D ! fCAlg k (X n ) /O Xn denote the corresponding coCartesian fibration obtained via the unstraightening construction. Notice that pullback along O ! O Xn induces a functor g

(

  Base change of (3.4.4.3) along the morphism ⌧ n ↵ 1 x O Yx ! O Xn ) The zero derivations d 0 n,x in (3.4.4.3) assemble to give a a well defined functor d 0 n : K op ! D and similarly the d n,x induce a well defined functor d n : K op ! D. Denote 0 := F d 0 n and := F d n :

  Recovering the extension of the extension of the original diagram f 1 n+1 by means of the right adjoint G above) Notice that the rigidication of both (3.4.4.5) and (3.4.4.6) concides with the derivations d 0 and d displayed in (3.4.4.4), respectively. We can also consider the diagrams 0 and as morphisms 0 ! 0 and ! in the 1category Fun K op , fCAlg k (X n ) O Xn //O Xn . Thanks to [Lur09b, Proposition 3.3.3.2] we can lift both diagrams 0 and as functors K

Corollary 3 .

 3 4.4.13. Let S be the saturated class generated by those morphisms f : A ! B in CAlg adm k such that the induced map Spf f rig : Spf(B) rig ! Spf(A) rig is an equivalence in the 1-category of derived k-affinoid spaces dAfd. Then the rigidification functor ( ) rig : CAlg adm k op ! dAfd factors as CAlg adm k op [S 1] ! dAfd and the latter is an equivalence of 1-categories.

1

 1

  (F, G), as desired. Corollary .1.2.3. Let X 2 dAn and f : F ! G be a morphism Coh + (X), where G is of bounded cohomology, i.e. G 2 Coh b

  2]. Remark .2.0.2. Both the pregeometries T ét (k) and T an (k) are unramified, see [Lur11a, Proposition 4.1] and [PY16a, Corollary 3.11], respectively. Proposition .2.0.3. The pregeometry T ad (k ) is unramified.

3 . 1 . 11 :

 3111 Lemma .3.0.1. Let (F, G) : C ! D be an adjunction of presentable 1-categories. Suppose further that:(i) Any epimorphism in C is effective;

Notation 4 .

 4 2.0.1.

Definition 4 .

 4 2.0.2. Let X be an 1-topos. A T-structure on X is a functor O : T ! X which commutes with finite products, pullbacks along admissible morphisms and takes ⌧ -coverings in effective epimorphisms. We denote by Str T (X) the full subcategory of Fun T (T, X) spanned by T-structures. A T-structured 1-topos is a pair (X, O), where X is an 1-topos and O 2 Str T (X).We can assemble T-structured 1-topoi into an 1-category denoted Top R (T). We refer to [Lur11d, Definition 1.4.8] for the precise construction. The functor Fun(T, ) : Cat 1 ! Cat 1 restricts to a functor Fun(T, ) : Top R op ! Cat 1 ,

and O 2

 2 Str T (X). We say that an object of TopR (T) is a T-structured 1-topos. Definition 4.2.0.3. Let X be an 1-topos. A morphism of T-structures ↵ : O ! O 0 is said to be local if for every admissible morphism f : U ! V in T the diagram

Example 4 .

 4 2.0.4.

  induced by the (t)-completion. (iii) The functor ( )an : T disc (k) ! T an (k), induced by the analytification. (iv) The functor ( ) rig : T ad (k ) ! T an (k) induced by Raynaud's generic fiber construction (cf. [Bos14, Theorem 8.4.3]).

Theorem 4 .

 4 2.0.8 (cf. [Ant18a,Lur11b,PY16b] ). Derived formal Deligne-Mumford stackk -stacks and derived kanalytic spaces assemble into 1-categories, denoted respectively dfDM k and dAn k , which enjoy the following properties: (i) fiber products exist in both dfDM k and dAn k ; (ii) The constructions given in Theorem 4.2.0.4 induce full faithful embeddings from the categories of ordinary formal Deligne-Mumford stackk -stacks fDM k and of ordinary k-analytic spaces An k in dfDM k and dAn k , respectively.

  ( ) rig : T ad (k ) ! T an (k) induced by Raynaud's generic fiber functor induces Top R (T an (k)) ! Top R (T ad (k )). [Lur11d, Theorem 2.1.1] provides a right adjoint to this last functor, which we still denote ( ) rig : Top R (T ad (k )) ! Top R (T an (k)).

Theorem 4 .

 4 2.0.10 ( [Ant18a, Corollary 4.1.4, Proposition 4.1.6]). The functor ( ) rig : Top R (T ad (k )) ! Top R (T an (k)) enjoys the following properties: (i) it restricts to a functor ( ) rig : dfDM taft ! dAn k .

Lemma 4 . 3 . 1 . 2 .

 4312 Let X be a derived k -adic Deligne-Mumford stackstack. Then an almost perfect sheaf F 2 Coh +

  3.1.4 and Theorem 4.3.1.6 at our disposal, the question of the non-emptiness of FM(F) is essentially reduced to the the following: Theorem 4.3.1.7. Let dfSch taft,qcqs k denote the 1-category of derived k -adic schemes which are quasi-compact, quasi separated and topologically almost of finite presentation. Then the functor Coh + loc : dfSch taft,qcqs k op ! Cat st 1 is a hypercomplete sheaf for the formal Zariski topology. Proof. A standard descent argument reduces us to prove the following statement: let f • : U • ! X be a derived affine k -adic Zariski hypercovering. Then the canonical map

Corollary 4 . 3 . 1 . 8 .

 4318 Let X 2 dfSch taft k

Lemma 4 . 3 . 3 . 1 .

 4331 Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Then the functor ( ) rig : Ind(Coh + (X)) ! Ind(Coh + (X rig ))

Remark 4 . 3 . 3 . 5 .

 4335 Notice that Theorem 4.3.3.4 holds without no bounded conditions on the cohomological amplitude on the considered almost perfect complexes. The key ingredient is the fact that the morphism Spec k ,! Spec k is an open immersion. Compare with [?, Lemma 6.5.3.7].

Lemma 4 . 3 . 3 . 8 .

 4338 Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + (X). Then for any G 2 Coh + nil (X), one has Map Ind(Coh + (X)) (G, F loc ) ' 0.

Corollary 4 . 5 . 0 . 4 .Then for any F 2

 45042 Let f : X ! Y be a proper and flat map between derived analytic spaces. Let p : Z ! Y be any other map and consider the pullback square Coh + (X) the canonical map

Proof.

  Using Theorem 4.4.0.1, we find a flat formal model f : X ! Y. Choose a formal model p : Z ! Y for p : Z ! Y , and form the pullback square W

  Proof. We only need to check the hypotheses of [PY17b, Theorem 7.1]. The representability of the truncation is guaranteed by [CG16, Proposition 5.3.3]. The existence of the global analytic cotangent complex has been dealt with in Theorem 4.6.0.1. Convergence and infinitesimal cohesiveness are straightforward checks. The theorem follows.

Proof.

  The same proof of [PY18b, Theorem 8.6] applies. It is enough to observe that Corollaries 4.5.0.3 and 4.5.0.4 allow to prove Lemma 8.4 in loc. cit. by removing the assumption of Y ! S being locally of finite presentation.

Theorem 5 . 1 . 1 . 1 .(X) 2

 51112 The moduli functor LocSys `,n (X) : Afd op k ! S is representable by a k-analytic stack. More precisely, there exists a k-analytic space LocSys framed `,n An k together with a canonical smooth map q : LocSys framed `,n (X) ! LocSys `,n (X) which exhibits LocSys framed `,n

  Afd k := (Afd op k ) op . Remark 5.2.1.5. Let A 2 Afd op k denote a k-affinoid algebra. The given of a presentation of A of the form A ⇠ = khT 1 , . . . , T m i/I determines a formal model for A, i.e a p-complete k -adic algebra of topological finite presentation A 0 such thatA ' A 0 ⌦ k k.in the category of k-algebras. One can simply take A 0 to beA 0 := k hT 1 , . . . T m i/I \ k hT 1 , . . . T m i.Definition 5.2.1.6. Given a k-affinoid alge bra A we denote by M(A) the set of semi-multiplicative seminorms on A. Given x 2 M(A) we can associate it a (closed) prime ideal of A. Namely, it corresponds to the kernel of x : A ! R, ker(x) ✓ A. The fact that it defines a prime ideal of A follows from multiplicativity of x 2 M(A). Notation 5.2.1.7. We denote by H(x) the completion of the residue field Frac(A/p), where p := ker(x). The field H(x) possesses a canonical valuation, denoted | • | x , induced by the one on A and given a 2 A we denote by |a| x 2 R the evaluation of | • | x on the image of a in H(x).

Remark 5 . 2 . 2 . 3 .

 5223 Fix a formal model A 0 for A, as above. The topology on A 0 admits the family {⇡ n A 0 } n 1 as a fundamental family of open neighborhoods around 0 2 A 0 . Consequently, for k 0, we have a fundamental family of normal open subgroups

When A 0

 0 is a formal model for A, we denote by LocSys framed `,n (U, 1 , . . . , l )(A 0 ) the set of those (M 1 , . . . , M r ) 2 GL n (A 0 ) r such that the mod p reduction (M 1 , . . . , M r ) = Id, mod p Remark 5.2.2.10. Let U 2 J r , A 2 Afd op k and A 0 be a formal model for A. Then the set LocSys framed `,n (U, 1 , . . . , l )(A 0 ) 2 Set does not depend on the choice of the topological generators for U . More precisely, if ⌧ 1 , . . . ⌧ s 2 U \ F r denote a different choice of topological generators for U , we have a natural bijection of sets LocSys framed `,n

  topological generators of a finite index normal open subgroup of b F r , which we shall denote by U . We remark that U is a free profinite by the version of Nielsen-Schreier theorem for open subgroups of free profinite groups, see [RS08, Theorem 3.3.1]. By Theorem 5.2.2.13 we conclude that ( 1 (M 1 , . . . , M r ), . . . , l (M 1 , . . . , M r )) 2 GL n (A) r defines a continuous group homomorphism

Theorem 5 . 2 . 2 . 18 .A 2

 522182 For each r 1, the functor LocSys framed `,n ( b F r ) : Afd op k ! Set, given on objects by the formula, Afd k 7 ! {(M 1 , . . . , M r ) 2 GL n (A) : there exists i, | i (M 1 , . . . M r ) Id|  |p|} 2 Set, is representable by a (strict) k-analytic space.

U, 1 0

 1 corresponds to those (M 1 , . . . , M r ) 2 LocSys framed `,n U, 1 , . . . , l 0 (A) such that h(M 1 , . . . , M r ) = Id, for every h 2 H \ F r ⇢ H. Then we have an equivalence of fiber products, Z := Sp A ⇥ ,..., l 0 LocSys framed `,n G, U, 1 , . . . , l As every k-affinoid algebra is Noetherian, [Con08a, Theorem 1.1.5], we conclude that Z parametrizes points which determined by finitely many equations with coefficients in A 2 Afd op k , induced from the relations defining H inside b F r , (after choosing topological generators for b F r ). We conclude that Z is a closed subspace of Sp A and thus representable. The result now follows. Remark 5.2.2.20. Given G a profinite group as above there is a canonical action of the k-analytic group GL n on LocSys framed `,n (G) via conjugation. Furthermore, continuous representations of a group correspond precisely to the conjugacy classes of elements in LocSys framed `,n (G) under the action of GL an n .

Definition 5 . 2 . 3 . 1 .

 5231 A geometric context (C, ⌧, P) consists of an 1-site (C, ⌧), see [Lur09b, Definition 6.2.2.1], and a class P of morphisms in C verifying:(i) Every representable sheaf is a hypercomplete sheaf on (C, ⌧).

Notation 5 . 2 . 3 . 2 .

 5232 Let (C, ⌧) denote an 1-site. We denote by Shv(C, ⌧) the 1-category of sheaves on (C, ⌧). It can be realized as a presentable left localization of the 1-category of presheaves on C, PSh(C) := Fun C op , S .

Definition 5 . 2 . 3 . 4 .

 5234 We say that F 2 Shv(C, ⌧) is locally geometric if F can be written as an union of n-geometricstacks F = S i G i , for possible varying n, such that each G i is open in F , i.e.,after base change by representable objects the corresponding inclusion morphisms are open immersions.An important feature that one desires to be satisfied in a geometric context (C, ⌧, P) is the notion of closedness under ⌧ -descent. Definition 5.2.3.5. Let (C, ⌧) be an 1-site. The 1-category C is closed under ⌧ -descent if for any morphism F ! Y , where F, Y 2 Shv(C, ⌧) and Y is required to be representable and for any ⌧ -covering

Remark 5 . 4 . 1 . 5 .

 5415 The 1-categories S and Cat 1 are presentable. It follows by [Lur09a, Remark 3.1.7] that one has a fully faithful embedding Pro(S) ! Pro(Cat 1 ). The analogous statement the larger versions b S and \ Cat 1 holds by the same reasoning. Definition 5.4.1.6. Let A 2 CAlg ad k , we define Perf ad (X)(A) as the functor category

Remark 5 . 5 . 1 . 1 .

 5511 2.25] and [GH15, Definition 5.4.3] are equivalent in the Cartesian symmetric monoidal case: Recall that in [Lur12c, Definition 4.2.5.25] the notion of a enrichment is defined as a pseudenrichment p : C ⌦ ! LM ⌦ together with the requirement that for any two objects M, N 2 C m we have a functorial morphism space object Mor M, N 2 C a . Following a private communication with David Gepner, whenever C a is a Cartesian symmetric monoidal 1-category, both definitions [GH15, Definition 5.4.3] and [GH15, Definition 7.2.14] are equivalent. Thanks to this fact, we can consider the 1-category of C a -enriched 1-categories, Cat 1 Pro(S) as the full subcategory of Alg cat (C a ) spanned by those complete objects, in the sense of [GH15, Definition 4.3.1].

Proposition 5 . 6 . 1 . 4 .

 5614 Let X 2 Pro(S fc ) be a connected profinite space. Let A 2 CAlg ad k <1 . Consider a perfect module M 2 Perf(A ⌦ k k) admitting a formal model M 0 2 Coh + (A). The fiber of the functor

Proposition 5 . 6 . 1 . 7 .

 5617 The functor Perf p (X) : CAlg ad k <1 ! Cat 1 descends to a well defined functor Perf p (X) : dAfd <1 k op ! Cat 1

  Under the evaluation functor ev(⇤) f corresponds to an idempotent morphism e f : M ! M. We might not be able to lift e f but thanks to [Ant18b, Corollary A.2.3] and [Lur09b, Proposition 4.4.5.20] there exists a formal model M 0 2 Coh + (A) for M for which we can lift e f : M ! M and higher homotopy coherences associated to the diagram e f : Idem ! Perf(A⌦ k k) to a diagram f : Idem ! Perf(A) such that its rigidification coincides with e

  is such that its materialization factors asMat (Y ) ! BGL n ( (Z)) ,! BEnd( (Z))in the 1-category S. See [1, §4.3 and §4.4] for more details. Definition 6.3.1.3. [22, Notation 3.6.1] We shall denote Sh ét

  Definition 6.3.1.6. Let Y := lim m Y m 2 Pro (S). Given an integer n 0, we define the n-truncation of Y as ⌧ n (Y ) := lim m ⌧ n (Y m ) 2 Pro(S n ), i.e. we apply pointwise the truncation functor ⌧ n : S ! S to the diagram defining Y = lim m Y m 2 Pro(S). Ind(Pro(S)) Notation 6.3.1.7. Let ◆ : Afd ! dAfd Q `denote the canonical inclusion functor. Denote by t 0 RLocSys `,n (X) := RLocSys `,n (X) ◆, the restriction of RLocSys `,n (X) to Afd Q `.

Proposition 6 . 3 . 1 . 8 .

 6318 The canonical morphismp : LocSys framed `,n (X) ! t 0 RLocSys `,n (X),in the 1-category St Afd Q `, ⌧ ét which induces an equivalence of stacks LocSys `,n (X) ' t 0 RLocSys `,n (X).

Proof. 2 2 )

 22 The proof of[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] Theorem 4.5.8] applies. Notation 6.3.1.9. Let Z := (Z, O Z ) 2 dAn denote a derived Q `-analytic space and M 2 Mod O Z . In[29, §5] it was introduced the analytic square zero extension of Z by M as the derived Q `-analytic space Z[M ] := (Z, O Z M ) 2 dAn, where O Z M := ⌦ 1 an AnRing k (Z) /O Z denotes the trivial square zero extension of O Z by M . In this case, we have a natural composite O Z ! O Z M ! O Z (6.3.1.2) in the 1-category AnRing k (Z) /O Z which is naturally equivalent to the identity on O Z . We denote p Z,M : O Z M ! O Z the natural projection displayed in (6.3.1.Definition 6.3.1.10.

A 2 1 )

 21 CAlg sm /l 7 ! Map Ind(Pro(S)) (BG, BEnd(A)) ⇥ Map Ind(Pro(S)) (BG,BEnd(A)) {⇢} 2 S. (6.4.1.Construction 6.4.1.1. [1, Proposition 4.2.6

Notation 6 . 4 . 1 . 2 .

 6412 Denote by L unr := Frac (W (l)) the field of fractions of W (l). It corresponds to the maximal unramified extension of Q `contained in L. Proposition 6.4.1.3. Let t 0 (Def ⇢ ) denote the 0-truncation of the derived formal moduli problem Def ⇢ , i.e. the restriction of Def ⇢ to the full subcategory of ordinary Artinian rings augmented over l, CAlg sm,/ l ✓ CAlg sm /l . Then t 0 (Def ⇢ ) is equivalent to Mazur's deformation functor introduced in [24, Section 1.2] and ⇡ 0 (A ⇢ ) is equivalent to Mazur's universal deformation ring. Proof. Given R 2 CAlg sm,/ l ✓ CAlg sm /l an ordinary (Artinian) local l-algebra, the object BEnd(R) 2 Ind(Pro(S)) is 1-truncated. Therefore one has a natural equivalence of spaces t 0 (Def ⇢ ) (R) ' Map Ind(Pro(S)) B⇡ ét 1 (X), BEnd(A) ⇥ Def⇢(k) {⇢}. (6.4.1.3)

  := A 0 b ⌦ A . . . b ⌦ A A 0 denotes the n + 1-tensor fold of A 0 with itself over A computed in the 1-category of derived adic W (k)-algebras CAlg ad W (k) . For a fixed [n] 2 op we an equivalence of spaces Map CAlg ad

  .5.1.2) in the 1-category Mod Q `. (6.5.1.1) together with base change of (6.5.1.2) along the morphism Q `! (Z) provides us with a natural morphism Map Perf ad `(X )( (Z)) (1, 1) ! (Z)[ 2d]. By naturality of the previous constructions, we obtain a morphism pre-orientation

A 2 Given A 2

 22 CAlg(C), we can consider the 1-category of A-modules A-Mod(C) := LMod A (C), formally defined as the fiber product {A} ⇥ CAlg(C) LMod(C). Similarly, we define the 1-category of coalgebras in C as CoAlg(C) := CAlg(Cop)op. CoAlg(C) we set A-CoMod(C) := (A-Modop)op.

Lemma 7 . 2 . 1 . 1 .

 7211 Let C ⌦ be a symmetric monoidal 1-category. Let A be a bialgebra in C. Then the 1-category A-Mod admits a symmetric monoidal structure such that the forgetful functorA-Mod(C) ! C is strong monoidal.

1 A

 1 ⌦c where is the coaction of k[⌘] on A and c : k[⌘] ! k is the counit of k. We now remark that the algebra structure on k[⌘] is the one induced by the coalgebra structure of k["]. Unraveling the definitions, we see that ⌘ 2 = 0 and therefore we can identify k[⌘] with ⌦ 1 (k[ 1]), the (nonconnective) split square-zero extension of k by k[ 1]. We claim that A ⌦ k k[⌘] can be canonically identified with the split square-zero extension A A[ 1]. Notation 7.2.1.4. Let f : A ! B be a morphism in CAlg nc k . The operations of pullback and pushout along f induce an adjunction

  f

Corollary 7 . 2 . 1 . 8 .

 7218 The forgetful functor v ⌘ : ⌘-CAlg nc k ! CAlg nc k is monadic.

Definition 7 . 3 . 1 . 1 .

 7311 Let (T, adm, ⌧) be a pregeometry. A T-geometric context is the data of:(i) a full subcategory C ⇢ Top R (T)containing at least all the objects of the form Spec T (X) for X 2 T;

Example 7 . 3 . 1 . 2 .

 7312 (v) the triple (C, ⌧, P) forms a geometric context in the sense of [PY16d, §2.2]. The following are the fundamental examples considered in this paper: Let k be a classical commutative ring (of any characteristic). Let T = T ét (k) be the étale pregeometry (see [Lur11d, Definition 4.3.1]). Then we take C := dA↵ k , the 1-category of derived affine k-schemes. The topology ⌧ coincides with the étale topology, and we take P to be the collection of smooth morphisms in dA↵ k . Notice that in this example already the inclusion C ⇢ Top R (T) does not commute with finite limits. Example 7.3.1.3. Let T an (C) be the complex analytic pregeometry (see [Lur11b, Definition 11.1]). Then we take C := dStn C , the 1-category of derived Stein spaces. This is the full subcategory of dAn C spanned by those derived complex analytic spaces X whose truncation t 0 (X) is Stein (cf. [Por15b, Definition 3.2]). The topology ⌧ coincides with the analytic topology, and we take P to be the collection of smooth morphisms in dStn C . Example 7.3.1.4. Let k be a non-archimedean field equipped with a non-trivial valuation. Let T = T an (k) be the k-analytic pregeometry (see [PY16b, Construction 2.2]

Definition 7 . 3 . 2 . 1 .Remark 7 . 3 . 2 . 2 .

 73217322 Let O be a T nc -structure on an 1-topos X. Then we refer to U (O) as the underlying spectrum object of O. Loosely speaking, we can identify U (O) with the collection of objects {O(E n )} n 0 . Notice that the assumption on O guarantees that O(E n ) ' ⌦(O(E n+1

Example 7 . 3 . 2 . 3 .

 7323 Consider the case T = T ét (k) and let O be a nonconnective T-structure on S.

BRemark 7 . 3 . 2 . 4 .

 7324 (G a ) ⇥ B(G a ) ! B(G a ),although there are of course multiplication mapsB n (G a ) ⇥ B m (G a ) ! B n+m (G a ),corresponding to the cup product in cohomology. In the general case, we can roughly think of a nonconnective T-structure in S as the given of:(i) a spectrum A 2 Sp;

(iii) for each C 2 D

 2 the object Q(C) is an elemmaentary T-structure (see [Lur11b, Definition 2.6]);

Corollary 7 . 3 . 3 . 7 .

 7337 For any 1-topos X with enough points, the commutative diagramStr T nc (X)Str T

(

  A) : T disc (k) nc ! S by setting (A)(B n (G a )) := Map(Sym k (k[ n]), A). Notice that (A)(B n (G a )) is equivalent, as chain complex, to ⌧ 0 (A[n]). This implies that we can identify U ( (A)) with the underlying spectrum of A. In other words, the diagramcdga kStr T disc (k) nc (S) V denotes the forgetful functor.Theorem 7.3.4.1. The functor is an equivalence.

  k (k[ n]) can be identified with the global sections of B n (G a ), while the Yoneda lemmama allows to identifyF (k[ n]) with Map T nc (B n (G a ), ). Now, the adjunction O a Spec introduced in [Toë06a] implies that (Sym k (k[ n])) ' Map T nc (B n (G a ), ),thus completing the proof.

01 := ⇢ 1 0 2

 12 : Str T nc 0 (X) ! Str T nc 1 (X), 10 := ⇢ 0 1 : Str T nc 1 (X) ! Str T nc 0 (X),that allow to change spectrum. Definition 7.3.6.1. Let (T, adm, ⌧) be a pregeometry and let (C, ⌧, P) be a T-geometric context. Let E 0 , E 1 Sp(Geom(C, ⌧, P)) be such that (C, P, E i ) is a T-nonconnective context, for i = 0, 1. The change of spectrum functors are the pair of functors 01 and 10 introduced above.

Proposition 7 . 3 . 6 . 2 .

 7362 For i = 0, 1 the diagram is left adjointable.

Proposition 7 . 3 . 6 . 3 .

 7363 For i = 0, 1 the diagram Str T nc i furthermore that the left Kan extension functor along T nc i ,! T nc 01 takes T nc i -structures to T nc 01 -structures. Then the above diagram is right adjointable. Remark 7.3.6.4. The condition on T nc

Proposition 7 . 3 . 6 . 10 .⌘◆

 73610 Let (T, adm, ⌧) be a pregeometry and let (C, ⌧, P) be an ordinary T-geometric context. LetE 0 , E 1 2 Sp(Geom(C, ⌧, P)) be such that (C, P, E i ) is a T-nonconnective context for i = 0, 1.Suppose furthermore that there is an endomorphisma : E 0 0 ! E 0 0in Mon gp E1 (Geom(C, ⌧, P)) and an equivalence in Mon gp E1 (Geom(C, ⌧, P)). Then the left Kan extension along T nc 0 ,! T nc 01 takes T nc 0 -structures to T nc 01 -structures. Proof. Let us simply write Lan for the functor Fun(T nc 0 , X) ! Fun(T nc 01 , X) given by left Kan extension along T nc 0 ,! T nc 01 . Consider the following claim:Claim. For every n 0 the canonical mapLan(O)(E n 1 )is an equivalence in X.Assume this claim. Then, as filtered colimits commute with finite limits, we deduce immediately that Lan(O)(E n 1 ) ' ⌦Lan(O)(E n+1 1

a n : E n 0 ! E n 0 for every n 0 .

 000 for every n 0 (when n = 0, a 0 = a). Moreover, since the delooping functorB n : Mon gp E1 (Geom(C, ⌧, P)) ! Mon gp E1 (Geom(C, ⌧, P) n )is a left adjoint, it commutes with filtered colimits. As a consequence, we obtain canonical equivalences in Mon gp E1 (Geom(C, ⌧, P)) Let now X 2 T nc 0 be any object. We claim that the canonical mapMap Geom(C,⌧,P) (X, E n 1 ) ! colim N MapGeom(C,⌧,P) (X, E n 0 )

Corollary 7 . 3 . 6 . 11 .

 73611 Under the same assumptions of Theorem 7.3.6.10 and for any T nc 0 -structure A, the diagramSp(Str T nc 01 (X) / i (A) ) Sp(Str T nc i (X) /A ) Str T nc 01 (X) / i (A)

' ⇤ : Str loc T nc 1 (

 1 X) ! Str loc T nc 0 (X)is an equivalence of 1-categories.Proof. The transformation ' induces a fully faithful functor T nc 0 ! T nc 1 that we still denote by '. Consider the left Kan extension functorLan ' : Str loc T nc 0 (X) ! Fun(T nc 1 , X). Notice, since (C 0 , P 0 , E 0 ) is a T 0 -nonconnective context, for every X 2 T nc 1 the functor (T nc 1 ) X/ ⇥ T nc 0 T 0 ! (T nc 1 ) X/ ⇥ T nc 0 [Lur11d, Proposition 3.28] implies that Lan ' : Str loc T0 (X) ! Fun(T 1 , X)factors through the (non full) subcategory Str loc T1 (X) and that such factorization is an equivalence. It is now enough to remark that since ' :(C 0 , ⌧ 0 , P 0 ) ! (C 1 , ⌧ 1 , P 1 ) is a Morita equivalence and ' s (E 0 ) ' E 1 , for every O 2 Str loc T nc 0 (X) we have Lan ' (O)(E n 1 ) ' O(E n 0 ). This proves at the same time that Lan ' : Str loc T nc 0 (X) ! Fun(T nc 1 , X) factors through the (non full) subcategory Str loc

7. 4 . 1

 41 The analytic nonconnective contexts Let T an (k) denote the analytic pregeometry. See Theorem 7.3.1.3 for the C-analytic case and Theorem 7.3.1.4 for the k-analytic case. Let dAn k denote the 1-category of derived analytic spaces, as defined in [Lur11b, Definition 12.3] and in [PY16b, Definition 2.5]. We endow dAn k with the étale topology ⌧ ét . This is indeed the restriction of the ⌧ ét -topology on Top R (T ét (k)): see [Por15b, Lemma 3.4] for the C-analytic case and [PY16b, Theorem 5.4] for the k-analytic case. The notion of smooth morphism between derived analytic spaces has been introduced and studied in [PY17b, §5.6]. Let us denote it by P sm . Then (dAn k , ⌧ ét , P sm ) is a geometric context in the sense of [PY16d, §2]. The 1-category dAn k has finite limits in virtue of [Lur11b, Proposition 12.12] in the C-analytic case and of [PY16b, Proposition 6.2] in the k-analytic case. Finally, the inclusion T an (k) ! dAn k preserves products, as it is shown in [Lur11b, Lemma 12.14(5)] and in [PY16b, Proposition 6.2(v)]. We can summarize these considerations in the following result: Proposition 7.4.1.1. The choice of the étale topology and the collection of smooth morphisms make (dAn k , ⌧ ét , P sm ) into a T an (k)-geometric context.We can also define a variant T afd an (k) of T an (k) as follows: the object of T afd an (k) are smooth Stein spaces (resp. smooth k-affinoid spaces), while the notion of admissible morphism and the Grothendieck topology are left unchanged. It follows from [Lur11d, Proposition 3.2.8] that the inclusion T afd an (k) ,! T an (k) is a Morita equivalence of pregeometries.Following the convention introduced in [PY17b], we let Afd k denote the full subcatgory of An k spanned by Stein spaces (in the C-analytic case) or by k-affinoid spaces (in the k-analytic case). Furthermore, we let dAfd k denote the full subcategory of dAn k spanned by those derived analytic spaces whose truncation belongs to Afd k . Corollary 7.4.1.2. The étale topology and the collection of smooth morphisms make (dAfd k , ⌧ ét , P sm ) into a T afd an (k)-geometric context. Proof. It is enough to prove that dAfd k is closed under fiber products in dAn k . As the truncation functor commutes with fiber products, it is enough to prove that Afd k is closed under fiber products in An k . In the Canalytic case, this follows from[GR84, §1.4.4]. In the k-analytic case, this is a consequence of [Con08b, Exercise 2.2.3[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]].

  loc T é t (k) (X)Str loc Tan(k) (X).

Proposition 7 . 4 . 2 . 4 .

 7424 Let X be an 1-topos and let A 2 Str loc Tan(k) (X). The underlying algebra functors( ) alg : Str loc Tan(k) (X) /A ! Str loc T é t (k) (X) /A alg ( ) alg : Str loc T nc an (k) (X) /i(A) ! Str loc T nc an (k) (X) /i(A alg ) are conservative.

  T nc disc (k) (S) with cdga k the local nonconnective algebraic structure(E n alg ) = (B n (G a )) 2 Str T nc disc (k) (S) corresponds to Sym k (k[ n]). We denote it by S k (n).

Lemma 7 . 4 . 2 . 7 .

 7427 There is a natural equivalenceS k (n) an ' S an k (n).

Theorem 7 . 4 . 2 . 8 (

 7428 Van Est, Lazard). Let ⇢ 2 |k ⇥ |[{1} denote the converging radius of the exponential function. Then if r  ⇢ the morphism ⌘ n,r is an equivalence. Moreover, if r, r 0 2 |k ⇥ | [ {1} satisfy r < r 0  ⇢ the natural morphism S an k (n, r 0 ) ! S an k (n, r) is an equivalence in AnRing nc k .

khr 1 T 1

 11 , . . . , r 1 T n i ! khr 1 T 1 , . . . , r 1 T n+1 i is determined by the rule T

Let A 2

 2 AnRing nc k and setS A (n) := A alg ⌦ k S k (n), S an A (n) := A b ⌦ k S an k (n), S an A (n, r) := A b ⌦ k S an k (n, r).

  (S an k (1, r)) b ⌦ l ' khr 1 T 1 , . . . , r 1 T l i,and the Yoneda lemma allows to identify the categorical completed tensor product A b ⌦ k (S an k (1, r)) b ⌦ l with the usual completed tensor product A b ⌦ k khr 1 T 1 , . . . , r 1

  defined informally by sending an 1-topos X to the presentable 1-category Str T nc an (k) (X), and a geometric morphism f ⇤ : X ⌧ Y : f 1 to the functorf ⇤ : Str T nc an (k) (X) ! Str T nc an (k) (X) obtained by composing with f ⇤ . The resulting functor dAn k Top R Pr R Str T nc an (k) ( )takes hypercoverings to limit diagrams. This is indeed a consequence of the descent theory for 1-topoi [Lur09c, 6.1.3.9] and [Por15b, Lemma 3.4] (in the C-analytic case) or [PY16b, Theorem 5.4] (in the k-analytic case). In particular, it extends to a functorStr T nc an (k) ( ) : dAnSt k ! Pr R . When X = (X, O X ) is a derived k-analytic space, we therefore abusively write Str T nc an (k) (X) instead of Str T nc an (k) (X). Proposition 7.4.3.1. Let X 2 dAnSt k and fix r 2 |k ⇥ | [ {1}. Consider the pullback square X ⇥ B(D m k (r)) B(D m k (r)) X Sp(k).

Map

  dAnSt k (Spec(k) an , B n (BG a )) ' Map dSt k (Spec(k), B n (G a )),which is tautological.Fix an 1-topos X. We are interested in studying the 1-category of S 1 -objects S 1 -Str loc

ank:

  Dop ! Str T nc an (k) (S) is strongly monoidal.Proof. We first observe that the functor an k is lax monoidal. Indeed, it is right adjoint to the functorAnSpec : Str T nc an (k) (S) ! dAnSt k op that sends A 2 Str T nc an (k) (S) to the derived analytic stack sending U 2 dAfd k to AnSpec(A)(U ) := Map Str T nc an (k) (A, an k (U )). The Yoneda lemma implies immediately that AnSpec is strong monoidal. It follows that its right adjoint is lax monoidal.Let now X, Y 2 D. Suppose at first that X, Y 2 T an (k). Then for any A 2 Str T nc an (k) (S), we haveA(X ⇥ Y ) ' A(X) ⇥ A(Y ).Applying the Yoneda lemma, we therefore obtainMap Str T nc an (k) ( an k (X ⇥ Y ), A) ' Map Str T nc an (k) ( an k (X), A) ⇥ MapStr T nc an (k) ( an k (Y ), A), and hence an k (X) b ⌦ k an k (Y ) ' an k (X ⇥ Y ).

p 1 :

 1 AnRing nc k ' Str loc T nc an (k) (S) ! Str loc T nc an (k) (X)is a left adjoint. This allows to identify an X (S 1) with an object in Comon E1 (Str loc T nc an (k) (X)). Proposition 7.4.5.6. Let X be an 1-topos. There is a canonical equivalenceS 1 -Str loc T nc an (k) (X) ' Comod an X (S 1 ) (Str loc

  4. Theorem 1.5.2.3 implies that Fun ECat1 Sh ét (X), Perf( ) : dAfd op k ! Cat 1 parametrizes continuous representations of Sh ét(X) with values in E 1 -monoid objects in ind-pro-endomorphisms spaces. That is to say, we take into account both the profinite structure on Sh ét (X) and the ind-pro-structure on End(M ). We define the moduli functor of perfect continuous `-adic representations of Sh ét

	Definition 1.5.2.5.

(X) as PerfSys `(X ) := ( ) ' Fun ECat1 Sh ét (X), Perf( ) which lives naturally in the 1-category Fun dAfd op k , S . Definition 1.5.2.6. Let RLocSys `,n (X) : dAfd op k ! S denote the substack of PerfSys `(X ) spanned by rank n objects, i.e.

  can be realized as an element in cyclic homology HC LocSys `,n (X) 2 Mod k . We obtain thus the following important result: Theorem 1.5.4.1. The derived k-analytic stack LocSys `,n (X) 2 dSt dAfd k , ⌧ ét , P sm admits a canonical shifted symplectic structure ! 2 HC LocSys `,n (X) of degree 2 2d. Moreover, given ⇢ 2 LocSys `,

n (X)(Z) with Z 2 dAfd k the underlying non-degenerate pairing is equivalent to the Poincaré pairing

  Theorem 1.7.0.2. There exists a rigidification functor ( ) rig : dfDM k ! dAn k which coincides with the usual Raynaud's rigidification functor, when restricted to discrete objects. Moreover, the functor ( ) rig : dfDM k ! dAn k is compatible with n-truncations.

	Theorem 1.7.0.4. The rigidification functor
	( ) rig	: dfDM k ! dAn k
	is a localization functor. More precisely, it induces an equivalence

Theorem 1.7.0.3. Let X 2 dAn k be a derived k-analytic space. Then there exists X 2 dfDM k a derived k -adic Deligne-Mumford stack such that one has an equivalence X rig ' X in the 1-category dAn k .

  Let f : A ! B be a morphism of derived rings. We say that f is étale if ⇡

	3.2.1 Functor of points approach
	Let k 2 CAlg denote a commutative ring or more generally a derived commutative ring.
	Definition 3.2.1.1. The 1-category of derived affine schemes over Spec k is defined as
	dA↵ := CAlg k	op	.
	Definition 3.2.1.2.

  Let (dA↵, ⌧ ét ) denote the étale Grothendieck site on the 1-category dA↵. Let X = Spec A 2 dA↵ we denote Shv ét (X) := Shv(dA↵ /X , T ét ) the 1-topos of étale sheaves on X. Remark 3.2.1.5. The 1-category Shv ét (X) can be realized as a presentable left localization of PShv(X) := Fun dA↵ /X , S given by localizing at the class of morphisms forcing étale descent on objects of PShv(X). In particular, the 1-category Shv ét (X) is presentable and indeed an 1-topos. We denote dSt A↵, ⌧ ét ✓ Shv ét A↵ the 1-category of derived stacks as the full subcategory spanned by those étale sheaves X 2 Shv ét (A↵) which are hypercomplete. Let dLocRing k denote the 1-category whose objects are pairs (X, O) where X is a topological space and O 2 CAlg(X) a local CAlg(X)-valued sheaf on X. We denote dSch k the full subcategory of dLocRing k spanned by those couples (X, O) satisfying:

	Notation 3.2.1.3. One can equip the 1-category dA↵ with the étale topology. We shall denote (dA↵, ⌧ ét ) the corresponding étale site. Definition 3.2.1.7. Let X be a topological space and O 2 CAlg(X) a CAlg-valued sheaf on X. We say that O is local if at every point x 2 X, the stalk O x 2 CAlg is a local derived ring, i.e. ⇡ 0 (O x ) is a local ordinary commutative ring. Notation 3.2.1.4. Definition 3.2.1.6. One can give a definition of the 1-category of derived schemes in terms of (derived) locally ringed spaces: Definition 3.2.1.8.

  Remark 3.3.1.8. Let A 2 CAlg k (X) O alg be as in Theorem 3.3.1.7. The natural morphism f A : A t ! (A) alg , is continuous if we equip both A and (A) alg with the adic topologies determined by the ideals {I n } n and {J n } n as in Theorem 3.3.1.7, respectively. In this case we can upgrade the morphism f A to a morphism in the 1-category ét ). Let A 2 CAlg k (X) /O , we say that A is strictly Henselian if it belongs to the essential image of the functor CAlg sh k (X) /O ! CAlg k (X) /O alg given on objects by the formula

	CAlg ad k (X) thanks to Theorem 3.3.1.4.
	Definition 3.3.1.9. Let (X, O) 2 Top
	A 2 CAlg sh k (X) /O 7 ! A alg	:= A ◆ 2 CAlg k (X) /O alg
	where ◆ : T disc (k ) ! T ét (k ) is the canonical transformation of pregeometries.
	Remark 3.3.1.10. Notice that the functor CAlg sh k (X) /O ! CAlg k (X) /O introduced in Theorem 3.3.1.9 is fully faithful. This follows from [Lur11c, Proposition 4.3.19, Remark 2.5.13] together with [Lur09b, Proposition
	7.2.1.14] and the proof of [Por15a, Proposition 9.2]. Therefore, we will usually abusively consider CAlg sh k (X) /O
	as a full subcategory of CAlg k (X) /O .

R

(T

  /O ad ! fCAlg taft k (X) /O constructed in Theorem 3.3.2.3. Thanks to Theorem 3.3.1.11 the composite ( ) ad ad is an equivalence when restricted to the subcategory C ✓ CAlg ad,taft k (X) spanned by strictly Henselian objects. Therefore the left adjoint functor

	ad

O ad is fully faithful and its essential image coincides precisely with the full subcategory of CAlg ad k (X) /O ad spanned by those strictly henselian A 2 CAlg ad k (X) /O ad topologically almost of finite presentation. Proof. Consider the adjunction ad , ( ) ad : CAlg ad,taft k (X)

  1 an : Mod O ! AnRing k (X) /O . We shall denote O M := ⌦ 1 an (M ) and refer to it as the analytic split square zero extension of O by M . The functor ⌦ 1 an admits a left adjoint ⌃ 1 an : AnRing k (X) /O ! Mod O . Suppose we are given A 2 AnRing k (X) /O and consider the 1-category AnRing k (X) A//O . We can consider the analytic derivations functor Der an A (O, ) : Mod O ! S given on objects by the formula M 2 Mod O 7 ! Map AnRing k (X) A//O (O, O M ) . Such functor is corepresentable by the analytic cotangent complex relative to A ! O, which we denote by L an O/A . Explicitly, one has a natural equivalence of mapping spaces

  ). It is a derived k -adic Deligne-Mumford stack and by construction it is admissible. Both X rig and t n+1 (X) have equivalent underlying 1-topoi. The rigidification functor ( ) rig : fCAlg k (Z) /O0 ! AnRing k (X) /O commutes with finite limits. Thus the diagram (3.4.4.2) remains a pullback diagram after rigidification. For this reason, we obtain a canonical morphism

  1.1.1. Let Cat Ex 1 denote the 1-category of small stable 1-categories and exact functors between them.

	Proposition .1.1.2. [HPV16a, Theorem B.2] Let C be a stable 1-category and A ,! C a full stable subcategory. Then the pushout diagram
	A	C
	0	D
	exists in the 1-category Cat Ex 1 .

Definition .1.1.3. Let A, C and D as in Theorem .1.1.2. We refer to D as the Verdier quotient of C by A.

  this concludes the proof. Corollary .1.2.2. Let X 2 dAn and f : F ! G be a morphism Coh + (X), where G is of bounded cohomology, i.e. G 2 Coh b (X). Suppose we are given a formal model X 2 dfSch taft k

  , since the rows of the diagram (4.3.2.2) are Verdier quotients, we conclude that the rows in the above diagram are fiber sequences. Therefore, the leftmost vertical arrow is an equivalence if and only if the rightmost one is. Lemma 4.3.2.2. The Beck-Chevalley transformation (4.3.2.1) is an equivalence whenever evaluated on objects in Coh ~(U).

  Proposition 4.3.2.3. The Beck-Chevalley transformation (4.3.2.3) is an equivalence when restricted to the full subcategory lim Coh ~(U • ) of lim Ind(Coh + (U • )). Proof. The discussion right after [PY16d, Corollary 8.6] allows us to identify the functor

.3.2.3)

  Lemma 4.3.3.3. Let X 2 dfSch taft k be a quasi-compact and quasi-separated derived k -adic scheme topologically almost of finite presentation. Let F 2 Coh + (X) and G 2 Coh + nil (X). Then

	of Ind(Coh +	Cat st 1 . Passing to ind-completions, we deduce that Ind(Coh + (X)). Applying [HPV16b, Lemma 2.5 and Remark 2.6] we conclude that Ind(Coh + (X rig )) is a Verdier quotient (X rig )) is an
	accessible localization of Ind(Coh +	(X)). As these categories are presentable, we deduce that the localization
	functor ( ) rig admits a fully faithful right adjoint, as desired.

Notation 4.3.3.2. Let X 2 dfDM k . Given F, G 2 Ind(Coh + (X))

we write Hom X (F, G) 2 Mod k for the k -enriched stable mapping space in Ind(Coh + (X)).

  Using [PY17b, Corollary 5.44], we can find analytic derivations d ↵ : (t n X)[F] ! t n X and d : (t n Y )[G] ! t n Y making the following cube

		d0
	(t n X)[F]		t n X
	d↵
	t n X	t n+1 X	fn
		fn+1	(4.4.0.1)
	d0
	(t n Y )[G]		t n Y
	d
	t n Y	t n+1 Y

commutative. Here d 0 denote the zero derivation and we set f n := t n (f ), f n+1 := t n+1 (f ). The derivations d ↵ and d correspond to morphisms ↵ : Lan t n X ! F and : Lan t n Y ! G, respectively. Moreover, the commutativity of the left side square in (4.4.0.1) is equivalent to the commutativity of

f ⇤ n Lan t n Y f ⇤ n G Lan t n X F f ⇤ n ↵ in Coh + (t n X). Notice that, since f is flat, the morphism f ⇤ n F ! G

is an equivalence. Using Theorem 4.2.0.15 and the induction hypothesis, we know that L ad Yn is a canonical formal model for Lan t n X . Using Theorem 4.3.3.10, we can therefore find a formal model : L ad Yn ! G for the map . We now set F := f ⇤ n G. Using Theorem 4.3.3.11, we can find m 2 N n and a formal model e ↵ : L ad Xn ! F for t m ↵ together with a homotopy making the diagram

  1 n T n i := { X i1,...,in a i1,...,in T i1 1 . . . T in n 2 k[[T 1 , . . . T n ]]| a i1,...,in r i1 1 . . . r in n ! 0}, whose multiplicative structure is induced by the multiplicative structure on the formal power series ring k[[T 1 , . . . T n ]].

	Definition 5.2.1.2. A k-affinoid algebra is a quotient of a Tate algebra khr 1 1 T 1 , . . . , r 1 n T n i by a finitely gener-ated ideal I.

  for any G 2 Pro(FinGrp).Remark 5.2.2.7. Given U 2 J r , the quotient group b F r /U ' is finite and therefore of finite presentation. It follows that U admits a finite family of generators 1 , . . . , l . Moreover, thanks to the Nielsen-Schreier theorem the group U is topologically finitely generated free profinite. Consider furthermore the dense group inclusion homomorphismF r ! b F r ,then U \ F r ! U is a discrete subgroup of U which is again dense in U . Therefore, we can assume without loss of generality that 1 , . . . , l 2 U \ F r .

	Notation 5.2.2.6. Let us fix J r a final family of normal open subgroups of finite index in b F r , i.e., such that we have a continuous group isomorphism, lim U 2Jr b F Notation 5.2.2.8. Let = Q i e nj i i F r be a general element of the profinite group b F r . Suppose furthermore we 2 b are given a group morphism

r /U ' b F r .

  Notation 5.2.2.11. Following Theorem 5.2.2.10 we will denote the set LocSys framed `,n (U, 1 , . . . , l ) simply by

	LocSys framed `,n

  Thus the latter is necessarily discrete. The result now follows, since we have that ⇢ 1 (V ) is an open subset in b F r . Remark 5.2.2.16. Let GL an n denote the analytification of the general linear group scheme GL n over Speck. Proposition 5.2.2.14 allows us to write LocSys framed (U, 1 , . . . , l ). Where, for each U 2 J r and 1 , . . . , l , LocSys framed `,n (U, 1 , . . . , l ) is given on the objects of Afd k by the formula, GL an n computed in the category An k . Consider the following cartesian diagram Set corresponds to the subfunctor of Sp B B whose value on k-affinoid algebra A consists of the set X 0 U, 1 ,..., l ,B (A) := {f : B ! A : for each i, | i (M 1 , . . . , M r ) Id|  |p|, in A}. Therefore, at the level of the points, the functor It is clear from our description, that this latter functor is representable by a Weierstrass subdomain of Sp B B. As (GL 0 n ) r is a (strict) k-affinoid space, it follows that LocSys framed `,n U, 1 , . . . , l is representable in the category Afd k , (consider in the above diagram with Sp B B = GL 0

			LocSys framed `,n	U, 1 , . . . , l , B
	Notation 5.2.2.15. We will denote
	LocSys framed `,n	( b F r )(A) :=	colim U 2Jr, 1 ,..., l	LocSys framed `,n
				`,n	( b F r ) as a union of subfunctors, LocSys framed `,n
	A 7 ! LocSys framed `,n
	Proof. Let GL 0 n = Sp	B khT ij i[ 1 det ] denote the closed unit disk of GL an n and
			LocSys framed `,n	U, 1 , . . . , l	0 2 An k
	denote the pullback of LocSys framed `,n
		LocSys framed `,n	U, 1 , . . . , l , B	0	Sp	B B
		LocSys framed `,n	U, 1 , . . . , l	0	(GL 0 n ) r
	where B 2 Afd op k is a k-affinoid algebra and
			(M 1 , . . . , M r ) 2 GL 0 n (B) r	= GL n (B 0	) r
	corresponds to a given morphism of k-analytic spaces Sp	B B ! (GL 0 n ) r . It follows that
		LocSys framed `,n

U, 1 , . . . , l (A)

⇠ = Hom cont b F r , GL n (A) . U, 1 , . . . , l (A) := {(M 1 , . . . , M r ) 2 GL n (A) r : such that for each i 2 [1, l], | i (M 1 , . . . , M r ) Id|  |⇡|}.

Lemma 5.2.2.17. The functor LocSys framed `,n (U, 1 , . . . , l ) as above, is representable by a (strict) k-analytic space. U, 1 , . . . , l along the inclusion morphism GL 0 n ,! U, 1 , . . . , l , B 0 2 Fun Afd op k , 0 parametrizes those points x 2 Sp B B such that, for each i, | (M 1 , . . . , M r ) Id|(x)  |p|(x)

  By the proof of our previous result it follows that the functor

	LocSys framed `,n	U, 1 , . . . , l ,! LocSys framed `,n	U 0	, ⌧ 1 , . . . , ⌧ s
		LocSys framed `,n	U, 1 , . . . , l : Afd op k ! Set
	is representable by a k-analytic subdomain of LocSys framed `,n	U 0	, ⌧ 1 , . . . , ⌧ s . Since we are interested in the repre-
	sentability of the space
	LocSys framed `,n	b F r ⇠ =	colim

U 2Jr,generators 1 ,... l LocSys framed `,n U, 1 , . . . , l , we need to check that the inclusions

  In order to show such condition on T, we notice first that that we can choose U 2 J r of sufficiently large finite index in b F U, 1 , . . . , l /(GL an n ) r . By the k-analytic structure on U, 1 , . . . , l 2 An k we conclude that we can take V 1 , . . . , V n k-affinoid subdomains of LocSys framed `,n U, 1 , . . . , l satisfying the above condition.

	LocSys framed `,n

r such that x 2 LocSys framed `,n U, 1 , . . . , l lies in its relative interior Int LocSys framed `,n

  Definition 5.2.3.9. Let G be a profinite group of topological finite presentation. We define the k-analytic stack of continuous representations of G St Afd k , ⌧ ét , P sm . Thanks to Theorem 5.2.2.19 we obtain the following important result: Theorem 5.2.3.10. Let G be a profinite group of topological finite presentation. Then the groupoid-valued functor The result is a direct consequence of Theorem 5.2.3.8 together with Theorem 5.2.2.19. Corollary 5.2.3.11. Let X be a smooth and proper scheme over an algebraically closed field. Then the k-analytic stack parametrizing continuous representations of ⇡ ét 1 (X) is representable by a geometric stack. Proof. It follows immediately by Theorem 5.2.3.10 together with the fact that under such assumptions on X its étale fundamental group ⇡ ét 1 (X) is topologically of finite generation. GL an n is a smooth group object in Shv(Afd op k , ⌧ ét ) and the corresponding geometric context satisfies descent we conclude by Theorem 5.2.3.8 that the quotient [LocSys framed

	LocSys `,n (G) := [LocSys framed `,n n ] 2 LocSys `,n (G) : Afd k ! S (G)/GL an
	is representable by a geometric stack.
	Proof. Remark 5.2.3.12. As LocSys framed `,n
	`,n	(G)/GL an

(G) is a representable object in the 1-category Shv(Afd op k , ⌧ ét ), n ] is representable by a geometric stack. Remark 5.2.3.13. The geometric stack LocSys `,n (G) is not, in general, a mapping stack. However the reader should think of it as a continuous version of the latter. It would thus be desirable to say that LocSys `,n (G) is equivalent to Map Ind(Pro(S)) BG, BGL an n ,

  where A[t] denotes the derived A-algebra obtained from A by freely adding a variable t in degree 0.

	k is p-complete. Thanks to [Lur16, Remark 8.1.2.4] we have an equivalence Remark 5.4.1.2. Suppose A 2 CAlg ad of derived k -algebras

  as profinite groups. Therefore, it is natural to consider the moduli stackPerfSys ` Sh ét (X) 2 dSt dAfd k , ⌧ étas a derived extension of the moduli LocSys `,n (X) 2 St Afd k , ⌧ ét , P sm .

	see [Lur09a, §3.6]. By construction, Sh ét a canonical identification	Sh ét (X) 2 Pro(S fc (X) 2 Pro(S fc ) classifies étale local systems on X. Moreover, we have ).
		⇡ 1 Sh ét	(X) ' ⇡ ét 1 (X)
	(X) denote the substack spanned by continuous p-adic ) with values in rank n free modules. Definition 5.7.2.1. Let RLocSys `,n (X) ✓ PerfSys ` Sh ét representations of Sh ét (X) 2 Pro(S fc
	Proposition 5.7.2.2. We have a canonical equivalence of stacks
	t 0 (RLocSys `,n (X)) ' LocSys `,n (X)
	in the 1-category St(Afd

k , ⌧ ét ).

  The formal spectrum Spf A ⇢ is locally admissible, see [2, Definition 3.1.1]. We can thus consider its rigidificiation introduced in [2, Proposition 3.1.2] which we denote by Def rig ⇢ := (Spf A ⇢ ) rig 2 dAn Q `. Notice that Def rig ⇢ is not necessarily derived affinoid. Let Z 2 dAfd Q `, [2, Corollary 4.4.13] implies that any given morphism

  Remark 6.5.2.4. A type GAGA theorem for reductive groups together with a theorem of B. Totaro, see[START_REF] Totaro | Hodge theory of classifying stacks[END_REF] Theorem 10.2], that the de Rham cohomology of the classifying stack GL an n coincides with `-adic cohomology Mod Q `, where BGL top n denotes the topological classifying stack associated to the general linear group GL n . In particular, we obtain a morphism

	C ⇤ dR BGL an n	' C ⇤ dR BGL top n
	in the 1-category

  .2.1.2) Theorem 7.2.1.3 guarantees that this diagram is vertically left adjointable. Denoting by L (resp. L " ) the left adjoint to U (resp. U " ), we obtain in particular the commutativity of the diagram

	"-CAlg nc k	CAlg nc k
	"-Mod k	Mod k .

  1 , which is easily checked to be grouplike. This allows us to consider once again the 1-category LMod S 1 (PSh(C)). Having fixed these notations, Theorem 7.2.2.4 has the following immediate corollary: Corollary 7.2.2.5. There exists a canonical 1-functor f : S 1 -PSh(C) ! LMod S 1 (PSh(C)) making the diagram

	S 1 -PSh(C)	f	LMod S 1 (PSh(C))
			(7.2.2.3)
	forget		forget
		PSh(C)

commutative. Moreover, f is an equivalence.

Proof. We have canonical equivalences

LMod S 1 (PSh(C)) ' Fun(Cop, LMod S 1 (S))

and

S 1 -PSh(C) ' Fun(Cop, S 1 -S).

The result is therefore a direct consequence of Theorem 7.2.2.4.

A rectification result in the general case Theorem 7.2.2.5 works for any 1-category C. Notice however that when applied to C = S it does not recover the statement of Theorem 7.2.2.4. Our goal is to formulate an analogue of Theorem 7.2.2.4 for a more general 1-category C.

  Mon gp E1 (C ⇥ ). As in general the functor ⌦ 1 C is only oplax monoidal, the diagram (7.2.2.4) is not horizontally left adjointable in general. Definition 7.2.2.6. Let C be a presentable 1-category. We say that C satisfies the condition (M) if the oplax monoidal functor ⌦ 1 Proof. As the functor ⌦ 1 C : S fin ! C is strong monoidal, it induces a commutative diagram

	Mon gp E1 (S fin,⇥	)		Mon gp E1 (C ⇥	)
	S fin			C
	Mon gp E1 (S fin,⇥	)	L 0	Mon gp E1 (C ⇥	)
	S fin		⌦1 C	C,

C : S fin ! C is strong monoidal. When C satisfies the condition (M) we have the following improved situation: Lemma 7.2.2.7. Suppose that C satisfies the condition (M). Then the diagram L ⌦1 C is commutative. In particular, there is an equivalence S 1 C ' S 1 ⌦ 1 C as objects of C.

  4.2.2.11] we can represent X and Y as functorsF X , F Y : op ⇥ 1 ! Csatisfying the conditions already described at the beginning of the proof of Theorem 7.2.2.4. Furthermore we can describe ⇢

⇤ (X) and ⇢ ⇤ (Y ) as the functors e F X , e F Y : op ⇥ 1 ! C whose restriction to op ⇥ {1} coincides with the simplicial presentation of S 1 and such that the diagrams e

  2.3.1) are equivalences. It follows from Theorem 7.2.3.2 that both maps Sym k

  Proposition 7.4.5.2. Let ⇢ 2 |k ⇥ | [ {1} be the converging radius of the exponential function. If r  ⇢, then ⇡ r is an equivalence.Proof. We know from Theorem 7.4.2.4 that the underlying algebra functor ( ) alg is conservative. It is therefore enough to prove that ⇡ alg r is an equivalence. Thanks to Theorem 7.4.2.8 and ??, we see that it suffices to prove that the canonical morphism

k (S 1

Equivalently, A is almost of finite presentation if ⇡ 0 (A) is of finite presentation and the cotangent complex L A/k is an almost perfect complex over A.

The 1-category dAn k also contains k-analytic Deligne-Mumford stackstacks.

By definition, X is a derived k-affinoid space if t 0 (X) is a k-affinoid space.

In the applications, C will be the opposite of the 1-category of nonconnective structures.

The assumption on the characteristic is truly necessary. When k = Fp, one can show that H 1 (Sym k (k[START_REF] António | Moduli of p-adic representations of a profinite group[END_REF])) is a countable direct sum of copies of Fp.

Notice that the adjoint functor theorem implies the existence of a left adjoint to ⌧ 0 . What is no longer clear, however, is that this left adjoint is fully faithful.

The reader might wonder why bother working with Tan(k)-structures rather than with Tan(k) 0 -structures, since the results for the latter are nicer. The reason is that the analytification functor only defines a transformation of pregeometries T ét (k) ! Tan(k), and this transformation of pregeometries is crucial in derived k-analytic geometry.

Acknowledgments

Acknowledgments We are grateful to B. Hennion, M. Robalo, B. Toën, G. Vezzosi and T. Y. Yu for useful discussions related to the content of this paper.

We are especially grateful to B. Toën for inviting the second author to Toulouse during the fall 2017, when the bulk of this research has been carried out and to IRMA, Université de Strasbourg, for inviting the first author to Strasbourg during the spring 2018.

4.2 Preliminaries on derived formal and derived non-archimedean geometries

We can therefore use Theorem 4.2.0.12 to obtain canonical equivalences

Under these identifications, the functor ( ) rig becomes equivalent to the base change functor

Moreover, it follows from [Ant18a, Proposition A. 1.4] that there is a canonical identification

In particular, ( ) rig : Coh + (X) ! Coh + (X) is t-exact. The conclusion is now straightforward. ). We say that F 2 Coh + (X) is a formal model for F if there exists an equivalence F rig ' F in Coh + (X rig ). We let FM(F) denote the full subcategory of

spanned by formal models of F.

Our goal in this section is to study the structure of FM(F), and in particular to establish that it is non-empty and filtered when X is a quasi-compact and quasi-separated derived k -adic scheme. Notice that saying that FM(F) is non-empty for every choice of F 2 Coh + (X) is equivalent to asserting that the functor (4.1.0.1)

is essentially surjective. Proof. We let A := (X, O X alg) , B := (X rig , O X rig ). Then as in the proof of Theorem 4.3.1.2, we have identifications Coh + (X) ' Coh + (A) and Coh + (X rig ) ' Coh + (B), and under these identifications the functor ( ) rig becomes equivalent to

As B ' A ⌦ k k, we see that A ! B is a Zariski open immersion. The conclusion now follows from [HPV16b, Theorem 2.12].

To complete the proof of the non-emptiness of FM(F), it would be enough to know that the essential image of the functor Coh + (X) ! Coh + (X rig ) satisfies descent. This is analogous to [HPV16b,Theorem 7.3].

Definition 4.3.1.5. Let X be a derived k -adic Deligne-Mumford stackstack locally topologically almost of finite presentation. We define the stable 1-category Coh + loc (X) of m-local almost perfect complexes as the cofiber

We denote by L : Coh + (X) ! Coh + loc (X) the canonical functor. We refer to L as the localization functor. We summarize below the formal properties of m-local almost perfect complexes: Proposition 4.3.1.6. Let X be a derived k -adic Deligne-Mumford stack-stack locally topologically almost of finite presentation. Then: (i) there exists a unique t-structure on the stable 1-category Coh + loc (X) having the property of making the localization functor L : Coh + (X) ! Coh + loc (X) t-exact.

Chapter 5

Moduli of continuous p-adic representations of a profinite group

Moduli of p-adic representations of a profinite group 123

Étale cohomology of perfect local systems

Let ⇡ : X ! SpecK denote the structural morphism. For each integer n 1, we have a canonical equivalence of 1-categories Shv(SpecK, Z/`nZ) ' Mod Z/`nZ .

We have a pullback functor p ⇤ : Shv ét (SpecK, Z/`nZ) ! Shv ét (X, Z/`nZ), which associates to each Z/`nZ-module M the étale constant sheaf on X with values in M .

Proposition 5.3.1.1. Let X be a proper normal scheme over an algebraically closed field K. Then R (X ét , Z/`nZ) is a perfect complex of Z/`nZ-modules.

Proof. This is a direct consequence of the more general result [GL14, Proposition 4.2.15].

Definition 5.3.1.2. Let A be a derived ring. We say that A is Noetherian if it satisfies the following conditions:

(i) ⇡ 0 (A) is a Noetherian ring;

(ii) For each i 0, ⇡ i (A) is an ⇡ 0 (A)-module of finite type.

Definition 5.3.1.3. Let A be a derived ring and M 2 Mod A and A-module. We say that M has tor-amplitude  n if, for every discrete A-module N , (which can be automatically seen as a ⇡ 0 (A)), the homotopy groups

vanish for every integer i > n.

Lemma 5.3.1.4. Let A be a Noetherian simplicial ring and M 2 Mod A be an A-module such that ⇡ i (M ) ' 0 for sufficiently small i  0. Then M is a perfect A-module if and only if the following two conditions are satisfied:

(i) For each i, ⇡ i (M ) is of finite type over ⇡ 0 (A);

(ii) M is of finite Tor-dimension.

Proof. It is part of [Lur12c,Proposition 7.2.4.23].

Remark 5.3.1.5. Let A be a derived Z/`nZ-algebra and let N 2 Shv(X ét , A) be a local system of perfect Amodules on X ét . Thanks to [GL14, Proposition 4.2.2] it follows that N can be written as a (finite sequence) of retracts of (f V ) ! (A) 2 Shv(X ét , A), where (f V ) ! : Shv(V ét , Z/`nZ) ! Shv(X ét , Z/`nZ), denotes the exceptional direct image functor associated to an étale map f V : V ! X. Lemma 5.3.1.6 (Projection Formula). Let X be a scheme over an algebraically closed field K. Let A be a simplicial ring and let F 2 Shv ét (X, A). Then, for any M 2 Mod A we have a natural equivalence,

in the derived 1-category Mod A , where ⇡ denotes the structural morphism ⇡ : X ! Spec K.

Proof. Let C ⇢ Mod A be the full subcategory spanned by those A-modules M such that there exists a canonical equivalence R (X ét , M) ' R (X ét , A) ⌦ A M . It is clear that A 2 C and C is closed under small colimits as both tensor product and the direct image functor ⇡ ⇤ commute with small colimits. Consequently, by the fact that the 1-category Mod Z/`nZ is compactly generated (under small colimits) by the object A, the result follows.

have homotopy groups which are finitely generated ⇡ 0 (A)-modules. Therefore, as R (X ét , ) is an exact functor the assertion also follows for ⌧ n+1 N . We are thus dealt with condition (i) in Theorem 5.3.1. [START_REF] Bommel | The Grothendieck monodromy theorem[END_REF] We have another fiber sequence of the form, ⌧ >n N ! N ! ⌧ n N. As X is of finite cohomological dimension it follows by [GL14, Lemma 2.2.4.1] that for any given integer i there is a sufficiently large integer n such that ⇡ i (R (X ét , ⌧ >n N )) 2 Mod ⇡0(A) vanishes. By exactness of the global sections functor R it implies that ⇡ i (R (X ét , N)) and ⇡ i (R (X ét , ⌧ n N )) agree for sufficiently large n.

As X is of finite cohomological dimension we conclude that given M a discrete ⇡ 0 (A)-module, the A-module ⇡ ⇤ ⇡ ⇤ (M ) has non-zero homotopy groups lying in a finite set of indices. Using the projection formula we conclude once more that R (X ét , N) ⌦ A M ' R (X ét , N ⌦ A M ) can be obtained by a finite sequence of retracts of the A-module ⇡ ⇤ ⇡ ⇤ (M ). Consequently, under our hypothesis on M , it follows that ⇡ i (R (X ét , N) ⌦ ⇡0(A) M ) ' 0 for large enough i. Thus we conclude that R (X ét , N) is of finite Tor-dimension as an A-module and thus a perfect A-module.

Pro-etale lisse sheaves on X ét

It follows by our hypothesis on X and [BS13, Lemma 7.4.10] that the pro-étale and étale fundamental groups of X agree henceforth it suffices to consider representations of the étale fundamental group of X, ⇡ ét 1 (X). Definition 5.3.2.1 (Noohi group). Let G be a topological group and consider the category of G-sets, denoted G-Set. Consider the forgetful functor F G : G-Set ! Set. We say that G is a Noohi group if there is a canonical equivalence G ' Aut(F G ), where Aut(F G ) is topologized with the compact-open topology on Aut(S) for each S 2 Set. Lemma 5.3.2.2. Let G be a topological group which admits an open Noohi subgroup U , then G is itself a Noohi group.

Proof. This is [BS13,Lemma 7.1.8].

Lemma 5.3.2.3. Let A be an k-affinoid algebra, then GL n (A) is a Noohi group.

Proof. Let A 0 be a formal model for A, it is a p-adically complete ring and we have the equivalence,

which induces its structure of topological group, in particular it is a pro-discrete group as in [Noo04, Definition 2.1]. Moreover, the system {GL n (A 0 )} [ {Id + p k M n (A 0 )} is a basis of open normal subgroups for the topology on GL n (A 0 ) and thus by [Noo04, Proposition 2.14] we conclude that GL n (A 0 ) is a Noohi group. As A 0 is an open subgroup of A the same holds for GL n (A 0 ) ⇢ GL n (A) and by [BS13,Lemma 7.1.8] we conclude that GL n (A) is a Noohi group.

The following Proposition is a generalization of [BS13,Lemma 7.4.7] and its proof is just an adaption of that one. We give it here for the sake of completeness. Proposition 5.3.2.4. Let A be a k-affinoid algebra. Then there is an equivalence of groupoids, LocSys `,n (X) ' Loc X,n (A), and Loc X,n (A) the groupoid of (pro-)étale local systems of rank n A-free modules on X.

Proof. Let X i 2 S /f and n 1 an integer. Each transition functor

is an exact functor between stable 1-categories, see [Lur12c,Proposition 1.1.4.6]. Thanks to [Lur12c,Theorem 5.5.3.18] the colimit colim

is again a stable 1-category, as the transition maps are exact. Furthermore, each 1-category Fun X i , Perf(A n ) admits a symmetric monoidal structure which is induced by the one on Perf(A n ) objectwise. Since the transition maps above are symmetric monoidal functors, one concludes that the 1-category in ?? is naturally endowed with a symmetric monoidal structure and by construction it is A n -linear. Each of the transition functors colim

are colimit preserving and exact. The fact that Perf ad (X)(A) is idempotent complete follows by stability of idempotent completion under filtered colimits [Lur09b,Propositin 4.4.5.21] and limits of 1-categories. Therefore, thanks to [Lur12c, Proposition 1.1.4.4] one deduces that the limit

is stable, as desired. The fact the 1-category displayed in (5.4.1.2) is symmetric monoidal follows from our previous considerations together with [Lur09b,Proposition 3.3.3.2]. By taking ind-completion one deduces that Mod ad (X)(A) is presentable. The statements for Coh + (X) ad (A) and Vect(X) ad (A) are similar but easier.

Remark 5.4.1.11. The 1-category Perf ad (X)(A) is rigid, i.e. every object in Perf ad (X)(A) is dualizable, as the tensor product is computed objectwise. We conclude that the conditions in [Lur16,Definition D.7.4.1] are verified, thus Mod ad (X)(A) is a locally rigid 1-category. Construction 5.4.1.12. We have a functor

given on objects by the formula

Thanks to [GHN15a, Lemma 6.2 and Example 6.3] the association

is functorial. Thus we can consider the composite

given on objects by the formula

Via straightening we obtain a coCartesian fibration

Given a cofiltered diagram h : I ! S we can consider the pullback diagram

Geometric properties of Perf ad (X)

In this §we prove that Perf ad (X) has a rich geometrical information, namely it satisfies hyper-descent, it is nilcomplete and cohesive and it admits a global k -adic cotangent complex. Definition 5.4.2.1. We equip the 1-category CAlg ad k with the étale topology. Denote by (CAlg ad k , ⌧ ét ) the corresponding étale 1-site. Let P sm denote the class of smooth morphisms in the 1-category CAlg ad k . The triple (CAlg ad k , ⌧ ét , P sm ) forms a geometric context, which we refer to as the k -adic geometric context. The 1-category of geometric stacks on (CAlg ad k , ⌧ ét , P sm ) is denoted as dSt(CAlg ad k , ⌧ ét , P sm ). Lemma 5.4.2.2. The pre-sheaf Perf ad (X) : CAlg ad k ! Cat 1 satisfies étale hyper-descent. Proof. Let A • : op ! CAlg ad k denote an hyper-covering of a given derived k -adic algebra A 2 CAlg ad k . We have thus an equivalence A ' lim

[n]2 op A [n] in the 1-category CAlg ad k . Let m 1 be an integer. Modding out by p m produces an étale hyper-covering A • m : op ! CAlg k m in the 1-site (CAlg k n , ⌧ ét ). Therefore, we have an equivalence

m in the 1-category CAlg k m . Therefore, we have a chain of equivalences of the form

where we used in a crucial way the fact that Perf : CAlg ad k ! Cat 1 satisfies étale hyper-descent. Proposition 5.4.2.3. The stack Perf ad (X) : CAlg ad k ! Cat 1 is cohesive and nilcomplete. Before proving Theorem 5.4.2.3 we prove first some preliminary results: Lemma 5.4.2.4. Let A 2 CAlg ad k be p-complete and M 2 Coh + (A), then we have a canonical equivalence

where

As M is eventually connective it follows that we have an equivalence

The functor ⌦ 1 ad being a right adjoint commutes with limits. As a consequence, we have a chain of equivalences of the form

in the 1-category Coh + (A). Moreover, the morphism A ! A n induces a canonical equivalence of functors

The result now follows by the fact that A M is p-complete together with a standard cofinality argument.

Proof of Theorem 5.4.2.3. We first treat cohesiveness of Perf ad (X). The pre-sheaf of perfect complexes

is cohesive thanks to [Lur12a, Proposition 3.4.10] together with Theorem 5.4.2.4. As both filtered colimits and limits commute with fiber products, we deduce that the same is true for Perf ad (X). We now prove that Perf ad (X) is nilcomplete. Fix an integer n 1, the pre-sheaf of perfect modules Proposition 3.4.10]. This implies that given A n 2 CAlg k n we have natural equivalences preserves cofiltered limits, where the latter denotes the 1-category of pre-sheaves on Cat op 1 . Therefore, cofiltered limits in Pro(Cat 1 ) can be computed objectwise. Thus given A 2 CAlg ad k it follows that we have natural equivalences

as the functor ⌧ m : CAlg k ! CAlg k is a left adjoint and therefore commutes with pushouts, thus

Remark 5.4.2.5. The above result holds true with an analogous proof for the functors Vect ad (X) and Coh + (X) ad . However, the result does not hold for Mod ad (X) as it is not true already in the discrete case.

We now devote ourselves to the computation of a cotangent complex for Perf ad (X). We will need a few preliminary results first. Proposition 5.4.2.6. Let A 2 CAlg ad k and M 2 Coh + (A) which we assume furthermore to be p-torsion free. Let ⇢ 2 Perf ad (X)(A). Given ⇢ 0 2 Perf ad (X)(A M ) together with a morphism

in the 1-category Coh + (X) ad (A), which we assume to be an equivalence after base change along the canonical morphism

Proof. It suffices to prove the result in the case where

in the 1-category Perf ad (X)(A M ) coincides with the identify morphism

) and its image in Coh + (X) ad (A) is equivalent to the zero object. We wish to prove that the coevaluation morphism

is the zero map in the 1-category Coh + (X) ad (A). Consider the inclusion morphism

in Mod(A). By naturality of taking tensor products we obtain that tensoring the above morphism with the coevaluation morphism induces a commutative diagram of the form

Since coev ⌦ A corresponds to the coevaluation morphism of the dualizable object

it coincides with the identity morphism

which is the zero morphism, by our assumption on ↵. It follows that the A-linear morphism

is the zero morphism, and thus by adjunction (with respect to the extension and restriction of scalars along A ! A M ), the coevaluation map coev :

is the zero morphism, as desired. Thus cofib(↵) ' 0 in the 1-category Coh + (X) ad (A M ), as desired.

can be identified canonically with

Proof. The canonical functor p A,M exhibits Perf ad (X)(A M ) as an object in the 1-category S /Perf ad (X)(A) ad . This is a right fibration of spaces and thus it induces a functor

Given ⇢ 2 Perf ad (X)(A) we have an equivalence of spaces

Thanks to Theorem 5.4.2.6 and its proof the loop space based at the identify of the object

Construction 5.5.1.3. Let Mat : Pro(S) ! S denote the materialization functor given on objects by the formula X 2 Pro(S) 7 ! Map Pro(S) ⇤, X The functor Mat preserves limits and it thus lifts to a symmetric monoidal functor

where we consider the corresponding Cartesian symmetric monoidal structures on both 1-categories. Furthermore, thanks to [GH15, Corollary 5.7.6] we have a realization functor

We have an equivalence of 1-categories Cat 1 (S) ' Cat 1 , thanks to [GH15,Theorem 5.7.6]. Therefore, we obtain an induced functor

Similarly, we have a materialization functor Mat : Ind(Pro(S)) ! S given on objects by the formula

Ind(Pro(S)) ⇤, X 2 S. By construction, this functor commutes with finite limits. Therefore, we are given a well defined functor

Therefore, we can consider End(M ) naturally as an object in the 1-category Mon E1 Pro(S) . Remark 5.5.1.5. We have a canonical functor

induced by the universal property of the pro-construction together with the canonical inclusion functor S ,! Cat 1 . For this reason, given X 2 Pro(S) we can consider the 1-category of continuous k -adic representations of X defined as the functor 1-category

Then there exists a canonical functor ⇡ A : Fun Cat1(Pro(S)) X, Perf(A) ! Perf(A) which sends a continuous A-adic representation ⇢ : X ! Perf(A) to the underlying perfect A-module M := ⇢(⇤) 2 Perf(A).

Enriched 1-categories and p-adic continuous representations of homotopy types

in the 1-category Sp pro (p).

Remark 5.5.2.4. Let A 2 CAlg ad k be a derived k -adic algebra. For each n 1, the derived k n -algebra A n admits a natural action of S/p n . Therefore, the stable 1-category Perf(A n ) is not only enriched over spectra but actually enriched over the derived 1-category Mod S/p n . The existence of the symmetric monoidal functor displayed in (5.5.2.1) implies the existence of an induced action of the 1-category Sp pro (p) on Perf(A n ). Passing to the limit, we deduce that the 1-category Perf(A) can be upgraded naturally to an object in the 1-category Cat 1 Sp pro (p) .

Thanks to Theorem 5.5.2.4 we have two natural enriched structures on Perf(A), for A 2 CAlg ad k . Namely, a Pro(S)-enriched structure on Perf(A) and an Sp pro (p)-enriched structure. We will show that these are compatible in a sense which we will precise hereafter.

Remark 5.5.2.5. Consider the usual connective cover functor

It can be upgraded to a lax symmetric monoidal functor ⌦ 1,⌦ : Sp ⌦ ! S ⇥ , where S is considered with its Cartesian symmetric monoidal structure. Therefore, ⌦ 1 induces a lax symmetric monoidal functor

For this reason, there exists a natural functor ⌦

, and the result follows.

Lemma 5.5.2.9. Let M, M 0 2 Sp pro (p). Then we have an equivalence of mapping spaces

Proof. It is a direct consequence of the characterization of mapping spaces in 1-categories of ind-objects. (A), the induced functor

is both essentially surjective and fully faithful. By construction and [GH15, Definition 5.3.3] essential surjectiveness can be checked after applying the materialization functor

Furthermore, after applying Mat cat the functor displayed in (5.5.2.2) is equivalent to the canonical functor

which is essential surjective thanks to [Lur12a, Proposition 3.4.10]. Thus, we are reduced to show that the functor (A) such that their images in Coh + (A ⌦ k k) are perfect. We need to show that the functor displayed in (5.5.2.3) induces an equivalence

in the 1-category Sp pro (p) p 1 , where

The functor ✓ can be realized as the ind-localization at p of the map

in the 1-category Sp pro (p). It thus suffices to show that ✓ 0 is an equivalence in the 1-category Sp pro (p) after multiplying by a sufficiently large power of p. The 1-category Sp pro (p) is a stable 1-category, since it consists of pro-objects in the 1-category Mod S p nil . We are thus reduced to prove that cofib(✓ 0 ) is equivalent to the zero morphism in Sp pro (p) after multiplication by a sufficiently large power of p. Furthermore, we have an equivalence cofib(✓ 0

) ' lim

is the canonical morphism at the level of mapping spaces. The previous statement is a consequence of the dual statement concerning the commutation of filtered colimits with finite limits, see the proof of [Lur12c, Proposition 1. 1.3.6].

By assumption, both A and A 0 are m-truncated for a sufficiently large integer m > 0. It follows then that both M, N 2 Mod A and M 0 , N 0 2 Mod A 0 have non-zero homotopy groups concentrated in a finite number of degrees. We thus conclude that each mapping spectrum displayed in ?? has non-trivial homotopy groups living in a finite number of degrees, which do not depend on the integer n 0, and only at the special fiber at n = 1. Therefore, there exists a sufficiently large k > 0 such that each cofiber cofib(✓ 0 n ) is killed by p k , and k does not depend on the chosen n. Denote

We have a chain of equivalences

Each X n is killed by a certain power of p, we deduce that the same holds for Map (Mod S p ) nil Z i , X n 2 S. Such property is closed under filtered colimits, thus we conclude that colim i

Map

(Mod S p ) nil Z i , X n is of p-torsion. Therefore, since we have assumed A and A 0 to be truncated derived k -adic algebras, also the projective limit

is of p-torsion. We conclude thus that for any compact object Z 2 Ind Sp pro (p) the mapping space

is trivial. As a consequence we obtain that

in the 1-category Ind Sp pro (p) . This implies that ✓ is an equivalence, thus the functor displayed in (5.5.2.3) is fully faithful and thus an equivalence of 1-categories.

Then the mapping object

Proof. This is a direct consequence of Theorem 5.5.2.17.

Then we can regard End(M 0 ) 2 Mon E1 (S) with an enhanced pro-structure seen as the pro-object ) 2 Ind(Pro(S)).

Thanks to Theorem 5.5.2.18 it follows that the above definition does not depend on the choice of the perfect formal model M 0 2 Perf(A). Moreover, as filtered colimits commute with finite limits, when taking ind-completions, it follows that the monoid structure on End(M 0 ) 2 Mon E1 (Pro(S fc )) induces a monoid structure on End(M ) 2 Ind(Pro(S)). Thus we can consider End(M ) naturally as an object in the 1-category Mon E1 (Ind(Pro(S))). which we designate by the 1-category of continuous functors between C and D.

We have a canonical inclusion functor Ind(Pro(S)) ,! Cat 1 Ind(Pro(S)) . Given X 2 Pro(S fc ) we can consider it as an ind-pro-space via the composite

Pro(S fc

) ,! Pro(S) ,! Ind(Pro(S)).

Given A 2 CAlg ad k , we can thus consider the 1-category of continuous functors

) be a profinite space. Suppose further that X ' ⇤, then we have a canonical

) be connected profinite space. We have thus an unique, up to contractible indeterminacy, morphism

In order to show the existence of a cotangent complex for PerfSys `(X ) we will need the following technical result: Proposition 5.6.3.5. Let F 2 St(CAlg ad k , ⌧ ét , P sm ) and denote by F rig 2 St(dAfd k , , ⌧ ét , P sm ) its rigidification. Then if F admits an adic cotangent complex at a point

, then F rig admits a cotangent complex at the rigidification

which we denote by L an F rig ,x rig . Moreover, we have a canonical equivalence

Proof. The existence of L ad F,x implies that for every M 2 Coh + (A) we have functorial equivalences

Thanks to [Ant18b, Proposition A.1.4] the 1-category Coh + (X) is a Verdier quotient of Coh + (A) with respect to the full subcategory of torsion objects in the 1-category Coh + (A). Furthermore, it follows from [Ant18b, Proposition A.2.1] and its proof that the we have an adjunction

between presentable 1-categories where ( ) rig is an accessible localization functor and ( ) + is consequently fully faithful. We have an equivalence of mapping spaces

Since (M rig ) + 2 Ind(Coh + (A)) we can write it as a filtered colimit

where the last equivalence follows by the adjunction. Therefore, we can write

, where X := Spf(A) rig . We have a chain of equivalences

where both colimit indexing 1-categories are filtered and x 0 denotes the composite

This is justified as in the above colimit diagrams it suffices to consider only colimits indexed by the full subcategories of formal models for X and lying under A. Furthermore, we have an equivalence

where C denotes the 1-category of admissible formal models for X[N ]. This last assertion follows from the observation that a formal model for X[N ] consists of the given of an admissible formal model for X together with a formal model for N . Observe that filtered colimits commute with finite limits in the 1-category of spaces. Thus

We have thus an equivalence of the form

as desired. The result now follows from the observation that the right hand side of (5.6.3.1) is an invariant under hyper-descent.

Theorem 5.6.3.6. Let Z 2 dAfd k and M 2 Coh + (Z). Suppose we are given furthermore a morphism

then we have a canonical identification

in the 1-category Coh + (Z), where Mod p (X)(Z) := Ind Perf p (X)(Z).

Proof. We first observe that the derived Tate acyclicity theorem implies that we have a canonical equivalence of 1-categories

As in the proof of Theorem 5.4.2.7 we consider the right fibration of spaces

As the rigidification functor ( ) rig

: CAlg ad k ! dAn k preserves small extensions it follows that the statement of Theorem 5.4.2.6 still holds in this case. Therefore, reasoning as in the proof of Theorem 5.4.2.7 we obtain a chain of equivalences of (Z)-modules

) denotes the usual base change functor along the canonical functor

As PerfSys `(X ) is cohesive, it follows that we have a pullback diagram of the form

The conclusion now follows as in Theorem 5.4.2.7.

Corollary 5.6.3.7. Let X 2 Pro(S fc ) be a connected p-cohomologically compact and p-cohomologically perfect profinite space. Then for every p-complete Z 2 dAfd k and every ⇢ 2 PerfSys `(X )(Z) the functor

given on objects by the formula

Proof. The result is a direct consequence of Theorem 5.4.2.9 together with Theorem 5.6.3.5 whenever

is liftable. For a general ⇢ 2 PerfSys `(X )(Z) the result follows thanks to Theorem 5.6.2.8 as it implies that ⇢ is a retract of a liftable object.

Main results

Representability theorem

As we shall see, the moduli stack LocSys `,n (X) : Afd op k ! S admits a natural derived extension which it is representable with respect to the derived k-analytic context. Nonetheless, the moduli LocSys `,n (X) cannot be realized as a usual k-analytic space, instead it corresponds to a k-analytic stack. Therefore, one must show that the derived enhancement of LocSys `,n (X) is representable not by a derived k-analytic space but instead by a derived k-analytic stack. It would be thus desirable to have a representability type statement in the context of derived k-analytic geometry. Fortunately, such a result has been proved by M. Porta and T. Yu Yue in [PY17a]. As it will be of fundamental importance we shall motivate such result. Definition 5.7.1.1. Denote by (dAfd k , ⌧ ét , P sm ) the derived k-analytic geometric context where ⌧ ét denotes the étale topology on dAfd k and P sm denotes the class of smooth morphisms on dAfd k . Definition 5.7.1.2. Let F 2 dSt dAfd k , ⌧ ét be a stack. We say that F is a derived k-analytic stack if it is representable by a geometric stack with respect to (dAfd k , ⌧ ét , P sm ). (ii) The truncation t 0 (F ) 2 St(Afd k , ⌧ ét ) is geometric, F admits furthermore a cotangent complex and it is cohesive and nilcomplete.

Proof. Let A 2 Afd k , then RLocSys `,n (X)(Sp(A)) can be identified with the space

Therefore, by the universal property of 1-truncation we have a chain of equivalences

where the last equivalence follows from the fact that ⇡ ét 1 (X) is a group, therefore every morphism ⇡ ét 1 (X) ! End(A) should factor through the group of units of End(A) which coincides with GL n (A) with its k-analytic induced topology. The result now follows, by the fact that we can realize

Map

Ind(Pro(S)) B⇡ ét 1 (X), BGL n (A) with the geometric realization of the diagram . . .

and the fact that

X) we will denote RLocSys `,n (X) simply by LocSys `,n (X) from now on. Theorem 5.7.2.4. The moduli stack LocSys `,n (X) 2 St(dAfd k , ⌧ ét ) admits a cotangent complex. Given ⇢ 2 LocSys `,n (X)(Z) where Z 2 dAfd k is a derived k-affinoid space, we have an equivalence

where C ⇤ ét X, Ad(⇢) denotes the étale cohomology of X with coefficients in

Proof. Since X is smooth and proper it follows that Sh ét (X) is p-cohomologically compact and p-cohomologically perfect. Therefore, PerfSys `(X ) admits a cotangent complex and by restriction so does LocSys `,n (X). Moreover, the tangent complex of LocSys `,n (X) at the morphism

can be identified with the mapping space

. We are thus reduced to prove that

But this follows by the universal property of Sh ét (X) together with the fact that global sections of local systems with torsion coefficients on Sh ét (X) classify étale cohomology on X with torsion coefficients. The result follows now for liftable such ⇢ and for general ⇢ by Theorem 5.6.2.4. Proposition 5.7.2.5. The moduli stack LocSys `,n (X) is cohesive and nilcomplete.

Proof. This is a direct consequence of the analogous statement for PerfSys `(X ).

As a consequence we obtain our main result: Theorem 5.7.2.6. The moduli stack LocSys `,n (X) 2 dSt dAfd k , ⌧ ét is representable by a derived k-analytic stack.

Proof. The proof follows by the Representability theorem together with Theorem 5.7.2. Let X be a smooth scheme over an algebraically closed field k of positive characteristic p > 0. Without the properness assumption the étale homotopy group ⇡ ét 1 (X) fits in a short exact sequence of profinite groups

where ⇡ w 1 (X) and ⇡ tame 1 (X) denote the wild and tame fundamental groups of X, respectively. One can prove that the profinite group ⇡ tame 1 (X) is topologically of profinite type. However, the profinite group ⇡ ét 1 (X) is, in general, a profinite pro-p group satisfying no finiteness condition or whatsoever. Needless to say, the étale fundamental group ⇡ ét 1 (X) will in general not admit a finite number of topological generators. Consider X = A 1 k , the affine line. Its étale and wild fundamental groups agree, but they are not topologically of finite type.

For this reason, the main results of Theorem 6.2.3.4 do not apply for a general smooth scheme X. In particular, one cannot expect that the moduli of `-adic continuous representations of X, LocSys `,n (X), is representable by a Q `-analytic stack. The purpose of the current text, is to study certain moduli substacks of LocSys `,n parametrizing continuous representations

factors through a finite quotient p : ⇡ w 1 (X) ! . Denote LocSys `,n, (X) such stack. Our main result is the following: Theorem 6.1.1.1. The moduli stack LocSys `,n, (X) : Afd Q `! S can be promoted naturally to a derived moduli stack

In particular, Theorem 6.1.1.1 implies that the inclusion morphism of stacks

induces an equivalence on contangent complexes, in particular it is an étale morphism. We can thus regard RLocSys `,n, (X) as an admissible substack of RLocSys `,n , in the sense of Q `-analytic geometry.

The knowledge of the analytic cotangent complex allow us to have a better understanding of the local geometry of RLocSys `,n . In particular, given a continuous representation

Theorem 6.1.1.3. The derived moduli stack PerfSys `(X ) admits a natural shifted symplectic form !. Explicitly, given ⇢ 2 PerfSys `(X ) ! induces a non-degenerated pairing

which agrees with Poincaré duality.

By transport of strucure, the substack LocSys `,n (X) ,! PerfSys `(X ) can be equipped with a natural shifted sympletic structure. By restricting further, we equip the LocSys `,n, (X) with a shifted symplectic form ! .

Summary

Let us give a brief review of the contents of each section of the text. Both §2.1 and §2.2 are devoted to review the main aspects of ramification theory for local fields and smooth varities in positive characteristic. Our exposition is classical and we do not pretend to prove anything new in this context. In §2.3 we construct the (ordinary) moduli stack of continuous `-adic representations. Our construction follows directly the methods applied in [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. Given q : ⇡ w 1 (X) ! a continuous group homomorphism whose target is finite we construct the moduli stack LocSys `,n, (X) parametrizing `-adic continuous representations of ⇡ ét 1 (X) such that ⇢ |⇡ w 1 (X) factors through . We then show that LocSys `,n, is representable by a Q `-analytic stack (the analogue of an Artin stack in the context of Q `analytic geometry).

In §3, we show that both the Q `-analytic stacks LocSys `,n (X) and LocSys `,n, (X) can be given natural derived structures and we compute their corresponding cotangent complexes. It follows then by [START_REF] Porta | Representability theorem in derived analytic geometry[END_REF]Theorem 7.1] that LocSys `,n, (X) is representable by a derived Q `-analytic stack.

§4 is devoted to state and prove certain comparison results. We prove Theorem 6.1.1.2 and relate this result to the moduli of pseudo-representations introduced in [START_REF] Chenevier | The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings[END_REF].

Lastly, in §5 we study the existence of a shifted symplectic form on LocSys `,n (X). We state and prove Theorem 6.1.1.3 and analysize some of its applications.

Convention and Notations

Throught the text we will employ the following notations:

(i) Afd Q `and dAfd Q `denote the 1-categories of ordinary Q `-affinoid spaces and derived Q `-affinoid spaces, respectively.;

(ii) An Q `and dAn Q `denote the 1-categories of analytic Q `-spaces and derived Q `-analytic spaces, respectively;

(iii) We shall denote S the 1-category of spaces and Ind(Pro(S)) := Ind Pro S the 1-category of ind-proobjects on S.

(iv) Cat 1 denotes the 1-category of small 1-categories and ECat 1 the 1-category of Ind(Pro(S))-enriched 1-categories.

(v) Given a continuous representation ⇢, we shall denote Ad(⇢) := ⇢ ⌦ ⇢ _ the corresponding adjoint representation;

(vi) Given Z 2 Afd Q `we sometimes denote (Z) := (Z) the derived Q `-algebra of global sections of Z.

Acknowledgements

I am grateful to Jean-Baptiste Tessier and Bertrand Toën for many useful discussions and suggestions on the contents of the present text. I would also like to acknowledge le Séminaire Groupes Réductifs et formes automorphes for the invitation to expose many of my ideas about the subject. Definition 6.2.2.3. Let G be a profinite group and p a prime numbger, we say that G is quasi-p if G equals the subgroup generated by all p-Sylow subgroups of G.

Examples of quasi-2 finite groups include the symmetric groups S n , for n 2. Moreover, for each prime p, the group SL n (F p ) is quasi-p. Let X = A 1 k be the affine line over an algebraically closed field k of characteristic p > 0. We have the following result proved by Raynaud which was originally a conjecture of Abhyankar: Theorem 6.2.2.4. [7, Conjecture 10] Every finite quasi-p group can be realized as a quotient of ⇡ ét 1 (X). Remark 6.2.2.5. In the example of the affine line the infinite nature of ⇡ 1 (A 1 k ) arises as a phenomenon of the existence of étale coverings whose ramification at infinity can be as large as we desire. This phenomenon is special to the positive characteristic setting. Neverthless, we can prove that ⇡ ét 1 (X) admits a topologically finitely generated quotient which corresponds to the group of automorphisms of tamely ramified coverings. On the other hand, in the proper case every finite étale covering of X is everywhere unramified. Definition 6.2.2.6. Let X ,! X be a normal compactification of X, whose existence is guaranteed by [START_REF] Nagata | Imbedding of an abstract variety in a complete variety[END_REF]. Let f : Y ! X be a finite étale cover with connected source. We say that f is tamely ramified along the divisor D := X\X if every codimension-1 point x 2 D is tamely ramified in the corresponding extension field extension k(Y )/k(X). Proposition 6.2.2.7. Tamely ramified extensions along D := X\X of X are classified by a quotient ⇡ ét 1 (X) ! ⇡ t 1 (X, D), referred to as the tame fundamental group of X along D. Remark 6.2.2.8. Let X denote a smooth compactification of X and D := X\X. We denote by ⇡ w 1 (X, D), the wild fundamental group of X along D, the kernel of the continuous morphism ⇡ ét 1 (X) ! ⇡ t 1 (X). Definition 6.2.2.9. Assume X is a normal connected scheme over k.

(i) Let f : Y ! X be an étale covering. We say that f is divisor-tame if for every normal compactification X ,! X, f is tamely ramified along D = X\X.

(ii) Let f : Y ! X be an étale covering. We shall refer to f as curve-tame if for every smooth curve C over k and morphism g : C ! X, the base change Y ⇥ X C ! C is a tame covering of the curve C. Remark 6.2.2.10. In Theorem 6.2.2.9 X is assumed to be a normal connected scheme over a field of positive characteristic. Currently, we lack a resolution of singularities theorem in this setting. Therefore, a priori, one cannot expect that both divisor-tame and curve-tame notions agree in general. Indeed, one can expect many regular normal crossing compactifications of X to exist, or none.

Neverthless, one has the following result: Proposition 6.2.2.11. [17, Theorem 1.1] Let X be a smooth scheme over k and let f : Y ! X be an étale covering. Then f is divisor-tame if and only if it is curve-tame. Definition 6.2.2.12. The tame fundamental group ⇡ t 1 (X) is defined as the quotient of ⇡ ét 1 (X) by the normal closure of opens subgroup of ⇡ ét 1 (X) generated by the wild fundamental groups ⇡ w 1 (X, D) along D, for each normal compactification X ,! X. Remark 6.2.2.13. The notion of tameness is stable under arbitrary base changes between smooth schemes. In particular, given a morphism f : Y ! X between smooth schemes over k, one has a functorial well defined morphism

Moreover, the profinite group ⇡ t 1 (X) classifies tamely ramified étale coverings of X. Remark 6.2.2.14. The tame fundamental group ⇡ t 1 (X) classifies finite étale coverings f : X ! Y which are tamely ramified along any divisor at infinity. Definition 6.2.2.15. We define the wild fundamental group of X, denoted ⇡ w 1 (X), as the kernel of the surjection

It is an open normal subgroup of ⇡ ét 1 (X). Proposition 6.2.2.16. [START_REF] Clark | Fundamental Groups in Characteristic p[END_REF] Let C be a geometrically connected smooth curve over k. Then the wild fundamental group ⇡ w 1 (C) is a pro-p-group. Theorem 6.2.2.17. [5, Appendix 1, Theorem 1] Let X be a smooth and geometrically connected scheme over k. There exists a smooth, geometrically connected curve C/k together with a morphism f : C ! X of varieties such that the corresponding morphism at the level of fundamental groups

is surjective and it factors by a well defined morphism ⇡ t 1 (C) ! ⇡ t 1 (X). In particular, ⇡ t 1 (X) is topologically finitely generated. Remark 6.2.2.18. Theorem 6.2.2.17 implies that ⇡ t 1 (A 1 k ) admits a finite number of topological generators. In fact, the group ⇡ t 1 (A 1 k ) is trivial.

Moduli of continuous `-adic representations

In this §, X denotes a smooth scheme over an algebraically closed field of positive characteristic p > 0. Nevertheless, our arguments apply when X is the spectrum of a local field of mixed characteristic.

Remark 6.2.3.1. Let A 2 Afd be Q `-affinoid algebra A 2 Afd. It admits a natural topology induced from a choice of a norm on A, compatible with the usual `-adic valuation on Q `. Given G an analytic Q `-group space we can consider the corresponding group of A-points on G, G(A). The group G(A) admits a natural topology induced from the non-archimedean topology on A. In the current text we will be interested in studying the moduli functor parametrizing continuous representations

Nevertheless, our arguments can be directly applied when we instead consider the moduli of continuous representations

By the results of the previous §, the étale fundamental group ⇡ ét 1 (X) is almost never topologically finitely generated in the non-proper case. For this reason, we cannot expect the functor LocSys framed `,n (G X ) to be representable by an object in the category An Q `of Q `-analytic spaces. Nevertheless, we can prove an analogue of Theorem 6.2.3.4 if we consider instead certain subfunctors of LocSys framed `,n . More specifically, given a finite quotient q : ⇡ w 1 (X) ! we can consider the moduli parametrizing continuous `-adic representations of ⇡ ét 1 (X) whose restriction to ⇡ w 1 (X) factors through : Construction 6.2.3.5. Let q : ⇡ w 1 (X) ! denote a surjective continuous group homomorphism, whose target is a finite group (equipped with the discrete topology). We define the functor of continuous group homomorphisms ⇡ ét 1 (X) to GL n ( ) with -bounded ramification at infinity, as the fiber product (X) introduced in Theorem 6.2.3.5 depends on the choice of the continuous surjective homomorphism q : P X ! . However, for notational convenience we drop the subscript q.

We have the following result: Theorem 6.2.3.7. The functor LocSys framed `,n, (X) is representable by a Q `-analytic stack.

Proof. Let r be a positive integer and denote F [r] a free profinite group on r topological generators. The finite group and the quotient G X /P X are topologically of finite generation. Therefore, it is possible to choose a continuous group homomorphism

, such that the images p(e i ), for i = 1, . . . , r, form a set of generators for , seen as a quotient of ⇡ w 1 (X), and for

).

Thanks to [ G we denote [F/G] the geometric realization of the simplicial object ] denote the moduli stack of rank n `-adic pro-étale local systems on X. Given a continuous surjective group homomorphism q : ⇡ w 1 (X) ! whose target is a finite group we define the substack of LocSys `,n (X) spanned by rank n `-adic pro-étale local systems on X ramified at infinity by level as the fiber product LocSys `,n, := LocSys `,n (X) ⇥ LocSys `,n (⇡ w 1 (X) LocSys `,n ( ) Theorem 6.2.3.15. The moduli stack LocSys `,n, (X) is representable by a Q `-analytic stack.

Proof. We have a canonical map LocSys framed `,n, (G X ) ! LocSys `,n, (X), which exhibits the former as a smooth atlas of the latter. The result now follows formally, as explained in [1, §2.3].

One can prove that there is an equivalence between the space of continuous representations

the space of rank n pro-étale A-local systems on X. We thus have the following statement: Proposition 6.2.3.16. [1, Corollary 3.2.5] The functor LocSys `,n (X) parametrizes pro-étale local systems of rank n on X.

Proof. The same proof of [1, Corollary 3.2.5] applies.

Derived structure

Let X be a smooth scheme over an algebraically closed field k and fix a finite quotient q : ⇡ w 1 (X) ! . In this §we will study at full the deformation theory of both the Q `-analytic moduli stacks LocSys `,n (X) and LocSys `,n, (X). Our goal is to show that LocSys `,n (X) and LocSys `,n, (X) can be naturally promoted to derived Q `-stacks, denoted RLocSys `,n (X) and RLocSys `,n, (X), respectively. Therefore the corresponding 0truncations t 0 RLocSys `,n (X) and t 0 RLocSys `,n, (X) are equivalent to LocSys `,n (X) and LocSys `,n, (X), respectively. We will prove moreover that both RLocSys `,n, (X) and LocSys `,n (X) admit tangent complexes and give a precise formula for these. Moreover, we show that the substack RLocSys `,n, (X) is geometric with respect to the geometric context dAfd Q `, ⌧ ét , P sm . In particular, RLocSys `,n, (X) admits a cotangent complex which we can understand at full.

We compute the corresponding cotangent complexes and analyze some consequences of the existence of derived structures on theses objects. We will use extensively the language of derived Q `-analytic geometry as developed in [START_REF] Porta | Derived non-archimedean analytic spaces[END_REF][START_REF] Porta | Representability theorem in derived analytic geometry[END_REF].

Derived enhancement of LocSys

Mat (X) := Map

Ind(Pro(S)) (⇤, X) 2 S, where ⇤ 2 Ind(Pro(S)) denotes the terminal object. This formula is functorial. For this reason, we have a well defined, up to contractible indeterminacy functor, materialization functor Mat : Ind(Pro(S)) ! S.

As a consequence of Theorem 6.3.1.1, there exists an object BEnd(Z) 2 Ind Pro S , functorial in Z 2 dAfd Q `, such that its materalization is equivalent to

The right hand side of (6.3.1.1) denotes the usual Bar-construction applied to E 1 -monoid object End( (Z)) 2 S. Moreover, given Y 2 Ind Pro(S) every continuous morphism

In particular, the counit of the adjunction ✓ : ◆ ⇤ ◆ ⇤ ! Id is an equivalence. Reasoning formally we deduce that ◆ ⇤ is fully faithful and therefore so it is g ⇤ . Definition 6.3.2.3. Let Sh tame (X) 2 Pro (S) denote the fundamental 1-groupoid associated to the 1-topos Shv(X tame ét ), which we refer to as the tame étale homotopy type of X. Remark 6.3.2.4. The fact that the geometric morphism g ⇤ : Shv(X tame ét ) ! Shv(X ét ) is fully faithful implies that the canonical morphism

induces an equivalence of profinite abelian groups ⇡ i Sh tame (X)

As a consequence one has a fiber sequence

in the 1-category Pro(S fc ) of profinite spaces.

Definition 6.3.2.5. The derived moduli stack of wild (pro)-étale rank n `-local systems on X is defined as the functor RLocSys w `,n (X) : dAfd op ! S given informally by the association

Remark 6.3.2.6. The functor RLocSys w `,n (X) satisfies descent with respect to the étale site (dAfd, ⌧ ét ), thus we can naturally consider RLocSys w `,n (X) as an object of the 1-category of derived stacks dSt (dAfd, ⌧ ét , ).

Suppose now we have a surjective continuous group homomorphism q : ⇡ w 1 (X) ! , where is a finite group. Such morphism induces a well defined morphism (up to contractible indeterminacy)

Precomposition along Bq induces a morphism of derived moduli stacks Bq ⇤ : RLocSys `,n ( ) ! RLocSys w `,n (X). Where RLocSys `,n ( ) : dAfd Q `! S is the functor informally defined by the association We can now give a reasonable definition of the moduli of local systems with bounded ramification at infinity: Definition 6.3.2.8. The derived moduli stack of derived étale local systems on X wtih -bounded ramification at infinity is defined as the fiber product RLocSys `,n, (X) := RLocSys `,n (X) ⇥ RLocSys w `,n (X) RLocSys `,n (B )

Proof. Let ⇧ := Bq : B⇡ w 1 (X) ! B denote the morphism of profinite homotopy types induced from a continuous surjective group homomorphism q : ⇡ w 1 (X) ! whose target is finite. We can form a fiber sequence

in the 1-category Pro S fc 1 ⇤/ of pointed 1-connective profinite spaces. Let A := (Z) and consider the 1categories Mod A (Sh w (X)) and Mod A (B ) introduced in Theorem 6.3.2.10. Let C A,n (B⇡ w 1 (X)) and C A,n (B ) denote the full subcategories of Mod A (B⇡ w 1 (X)) and Mod A (B ), respectively, spanned by modules rank n free A-modules. It is a direct consequence of the definitions that one has an equivalence of spaces

where ( ) ' denotes the underlying 1-groupoid functor. The fiber sequence displayed in (6.3.2.3) induces an equivalence of 1-categories

where the right hand side of (6.3.2.4) denotes the 1-category of Y-equivariant continuous representations of B⇡ w 1 (X) with A-coefficients. Thanks to [1, Proposition 4.4.9.] we have an equivalence of A-modules

and similarly,

By definition of ⇢, we have an equivalence ⇢ Y ' ⇢, where ( ) Y denotes (homotopy) fixed points with respect to the morphism Y ! B⇡ w 1 (X). Thus we obtain a natural equivalence of A-modules:

Homotopy Y-fixed points are computed by Y-indexed limits. As the Y-indexed limit computing the right hand side of (6.3.2.7) has identity transition morphisms we conclude that the right hand side of (6.3.2.7) is naturally equivalent to the mapping space

where ⇧ ⇤ : Mod A (B⇡ w 1 (X)) ! Mod A (B ) denotes a right adjoint to the forgetful ⇧ ⇤ : Mod A (B ) ! Mod A (B⇡ w 1 (X)). As a consequence we have an equivalence

in the 1-category S. Notice that, by construction

in the 1-category Mod A (B ). One has moreover equivalences

as the restriction of ⇢ ⌦ ⇢ _ to Y is trivial. Thanks to (6.3.2.5) through (6.3.2.11) we conclude that the canonical morphism LocSys `,n (B ) ! LocSys `,n (B⇡ w 1 (X)) induces an equivalence on tangent spaces, as desired. Construction 6.3.2.12. Fix a continuous surjective group homomorphism q : ⇡ w 1 (X) ! , whose target is finite. Denote by H the kernel of q. The profinite group H is an open subgroup of ⇡ w 1 (X). For this reason, there exists an open subgroup U  ⇡ ét 1 (X) such that U \ ⇡ w 1 (X) = H. In particular, the subgroup U has finite index in ⇡ ét 1 (X). As finite étale coverings of X are completely determined by finite continuous representations of ⇡ ét 1 (X), there exists a finite étale covering

1 (X) acts on it canonically. Moreover, one has an isomorphism of profinite groups of stacks.

Proof. By Galois descent, the restriction morphism along f U : Y U ! X induces an equivalence of stacks

Moreover, the considerations of Theorem ⇢ 2 RLocSys `,n, (X)(Z), the tangent complex at ⇢ T RLocSys `,n, (X),⇢ ' C ⇤ ét X, Ad(⇢) [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF] 2 Mod (Z) Definition 6.5.2.1. Let L RLocSys `,n (X) denote the cotangent complex of the derived moduli stack RLocSys `,n (X).

We will denote by

Remark 6.5.2.2. Notice that C ⇤ dR RLocSys `,n (X) admits, by construction, a natural mixed algebra structure. However, we will be mainly interested in the corresponding "plain module" and E 1 -algebra structures underlying the given mixed algebra structure on C ⇤ dR RLocSys `,n (X) . Proposition 6.5.2.3. Let X be a proper and smooth scheme over an algebraically closed field of positive characteristic p > 0. We then have a well defined canonical morphism

Proof. Let ⇢ 2 PerfSys `(X ) be a continuous representation. We have a canonical morphism

in the 1-category S, where ⇢(⇤) denotes the module underlying ⇢. This association induces a well defined, up to contractible indeterminacy, morphism

PerfSys `(X ) ! Perf an , where Perf an 2 dSt dAfd Q `, ⌧ ét denotes the analytification of the algebraic stack of perfect complexes, Perf. Therefore, we obtain a canonical morphism

Thus we can rewrite (6.5.2.1) simply as

As étale cohomology C ⇤ ét (X, Q `) 2 Mod Q `is a perfect module we can dualize (6.5.2.2) to obtain a canonical morphism By taking global sections in the above diagram we conclude that the composite

has support in Map S 1 , RLocSys `,n (X) ,! Map S 1 , PerfSys `(X ) . Therefore, we can factor the composite

Chapter 7

Analytic HKR theorems

Main results

The goal of this paper is to prove a highly structured version of the Hochschild-Kostant-Rosenberg theorem in the setting of analytic geometry. We do not wish to make any smoothness assumption, and this leads us to work with derived analytic geometry, as developed by J. Lurie, T. Y. Yu and the last author. In first approximation, the theorem we wish to prove is the following:

Theorem 7.1.0.1. Let k denote either the field C of complex numbers or a non-archimedean field of characteristic 0 with a non-trivial valuation. Let X be a k-analytic analytic space. Then there is an equivalence of derived analytic spaces X ⇥ X⇥X X ' TX [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF], compatible with the projection to X.

Suppose that X is Stein (when k = C) or affinoid. Let A := (X; O X ). Then the above theorem implies that there is the following equivalence of simplicial algebras:

where Sym an A denotes the analytification relative to A of the algebraic Sym A . From this point of view, we see that on the right hand side one has an extra structure that we ignored so far, namely the de Rham differential. Taking this extra structure into account leads to the following more precise version of the HKR theorem: Theorem 7.1.0.2. There are 1-categories "-AnRing k of mixed analytic rings and S 1 -AnRing k of S 1 -equivariant mixed analytic rings. These categories are equivalent compatibly with their forgetful functors to AnRing k .

Strategy of the proof

One novelty of this paper is the strategy itself that we use. Our method is new even in the algebraic case and provides an alternative proof of the main result of [TV11]. In order to explain our main ideas, we provide an axiomatic treatment of the HKR theorem.

Warning 7.1.0.3. In this axiomatic presentation, we formulate stronger hypotheses than what is actually needed. This is done in order to obtain a neater exposition. These extra assumptions will be satisfied in the algebraic and C-analytic setting, but not in the k-analytic one.

We start with an 1-category A, that plays the role of either CAlg k or AnRing k . Assumption 7.1.0.4. The 1-category A is presentable.

In particular, A has pushouts. We denote the pushout of the diagram

The functor

Der A (A; ) : A-Mod ! S given by sending M to Map A /A (A, A M ) commutes with limits and -filtered colimits for  a big enough regular cardinal. Therefore, it is representable by an object in A-Mod that we denote L A . Assumption 7.1.0.5. For every A 2 A there is a conservative functor

Furthermore, this functor admits a left adjoint, denoted Sym A A ( ).

Step 1: Construction of the categories of mixed and S 1 -equivariant objects. It is easy to construct the 1category of S 1 -equivariant objects in A. Indeed, we set

This category is equipped with a forgetful functor

which is conservative and admits both a left and a right adjoint. In particular, it is monadic. We denote by T S 1 the associated monad. Notice that we can identify the endofunctor of A underlying the monad with the functor

It is less trivial to construct the 1-category of mixed objects in A. In this general setting, we need an assumption: Assumption 7.1.0.6. There is an 1-category "-A equipped with a functor U " : "-A ! A satisfying the following properties:

(i) the functor U " is conservative, commutes with sifted colimits and it admits a left adjoint DR : A ! "-A.

(ii) For every A 2 A, there is a canonical equivalence

In particular, U " exhibits "-A as monadic over A. We let T " denote the associated monad.

At this point, we can distinguish two versions of the HKR theorem:

(i) The plain HKR: this is the statement that the underlying endofunctors of T S 1 and T " are equivalent. It implies the familiar algebraic formulation of the HKR theorem, i.e. the existence of an equivalence

(ii) The structured HKR: this is the statement that T S 1 and T " are equivalent as monads. It implies the existence of an equivalence of 1-categories "-A ' S 1 -A compatible with the forgetful functors U S 1 and U " to A.

Notice that the structured HKR implies the plain HKR, but the vice-versa is obviously not true.

Step 4: Contractibility of the space of coalgebra structures The final step required for the completion of the proof is to prove that the morphisms p : Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) ! A S 1 and q : Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) ! A " can be promoted to morphisms in CoAlg(A nc

). Even in the algebraic setting, this verification has never been done explicitly. The main theorem of [TV11] implies that A S 1 and A " are equivalent as bialgebras. However, in loc. cit. the authors do not give a direct argument, and therefore they are taking quite a long detour. One possible way of expressing the difficulty is the lack of a rectification theorem for bialgebras. With our approach it is possible to show directly that the morphism p respects the coalgebra structure. However, to verify that q also respects the coalgebra structure is a nontrivial task. Our method consists in verifying the following stronger statement: Assumption 7.1.0.16. The space of coalgebra structures on Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]), formally defined as the pullback

is contractible.

Pairing this assumption with Theorem 7.1.0.14 finally completes the proof of our main theorem. Let us be more specific about the way of checking this last assumption in practice. Once again, we need to unravel the actual construction of A nc and relate Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) with the cohomology of a classifying stack like B(BG a ). This allows to canonically identify the space of coalgebra structures on Sym nc k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) with the space of group structures on B(BG a ). In the cases of interest, we see B(BG a ) as an object in the 1-topos of derived (analytic) stacks. In particular, we are entitled to use the 1-categorical version of May's delooping theorem. This reduces the computation of the group structures on B(BG a ) to the computation of the space of E 1 -structures on BG a that are compatible with its additive structure. At this point, the Eckmann-Hilton argument implies that this latter space is contractible. See Theorem 7.2.3.2.

Structure of the paper

In Section 7.2 we briefly revisit the main theorems of [TV11] providing shortened proofs following our general strategy. We notice that in this case we have a natural candidate for the category A nc , namely the category of unbounded cdgas. Furthermore, Van Est theorem can easily be bypassed by means of a direct computation of Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). On the other hand the contractibility of the space of coalgebra structures on Sym k (k [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) (cf. Theorem 7.2.3.2) is a new result that was missing from both [BZN12] and [TV11].

In ?? we provide a general framework to produce the category A nc . In the algebraic setting, the category we obtain is bigger than the category of unbounded cdgas. It is nevertheless possible to canonically recover the category of unbounded cdgas out of our A nc . From the point of view of the HKR theorem, the distinction between the two categories is not relevant because Axioms (1) through (4) are satisfied in both cases.

In ?? we apply the machinery previously introduced to construct the 1-categories "-AnRing k , S 1 -AnRing k and the nonconnective variations. We conclude by proving the main theorem.

Revisiting the algebraic case

We start the paper by reviewing the algebraic setting for the HKR theorem. In this case, the machinery of nonconnective structures is not needed, as we have a natural candidate, namely the 1-category of unbounded cdgas. Nevertheless, we take the opportunity to collect a few basic facts about mixed algebras and S 1 -objects that are needed in what follows. In Section 7.2.1 we recast the theory of mixed algebras as developed in [TV11, ?, ?] in purely 1-categorical terms. We give two different description of this 1-category. The first one is equivalent in a more or less tautological way to the classical one introduced in the aforementioned papers, and it is "monadic" in nature. The second one is "comonadic" in nature and it has two main advantages: first of all, it makes the Proof. We observe that the 1-functor CAlg(C)op ! Cat 1 sending A to A-Mod(C) is lax monoidal. In particular, it brings an object A 2 CoAlg(CAlg(C)) to a symmetric monoidal 1-category A-Mod(C). Unraveling the definition, we see that the induced forgetful functor is strong monoidal.

Consider the 1-category "-Mod k of k["]-modules. We denote by ⌦ k the monoidal structure on "-Mod k provided by the previous lemmama. Definition 7.2.1.2. The 1-category of (nonconnective) mixed algebras is

By definition, "-CAlg nc k comes equipped with a forgetful functor

which is monadic. On the other hand the forgetful functor

is strong monoidal. In particular, it induces a functor

is vertically left adjointable. Furthermore, the functor v " commutes with all limits and colimits, and in particular it is both monadic and comonadic.

Proof. We first observe that since U , U " and u " are conservative, the same goes for v " . As both the 1-categories "-CAlg nc k and CAlg nc k are presentable, the fact that v " is monadic and comonadic follows at once if we prove that it commutes with both limits and colimits.

Observe that the functors U , U " and u " commute with limits and sifted colimits. As U is conservative, it follows that v " commutes with limits and sifted colimits as well. In order to prove that v " commutes with arbitrary colimits, it is therefore enough to prove that it commutes with arbitrary coproducts of free objects. This is a direct consequence of the vertical left adjointability of the diagram.

We are thus left to prove that the diagram is vertically left adjointable. The functors U " and U admit left adjoints L " and L. We have to prove that the Beck-Chevalley transformation

is an equivalence. Since U v " ' u " U " , we are reduced to check that the natural transformation U L u " ! u " U " L " is an equivalence. We now recall from [Lur12c,3.1.3.13] that there are canonical equivalences

The conclusion now follows from the fact that u " is strong monoidal and commutes with arbitrary colimits.

Proof. The functor f ⇤ : CAlg nc A//A ! CAlg nc B//B commutes with all limits. Taking its first Goodwillie derivative we obtain the above commutative diagram. In particular, for every M 2 A-Mod we have a Beck-Chevalley transformation

and we have to prove that it is an equivalence. We now observe that the underlying module of

On the other hand, the underlying module of ⌦ 1 (M ) is A M . Since the pushout in CAlg nc k is computed by the ordinary tensor product, we see that

As the forgetful functor B-Mod ! Mod k is conservative, we conclude that ↵ is an equivalence.

In particular, the coaction :

can be canonically identified with a derivation d of A with values in A [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. We refer to d as the derivation underlying the ⌘-algebra A. At this point, we can prove the following important result: Recall that

Using the universal property of the free algebra, we therefore obtain

On the other hand, composing with the canonical projection

we obtain a natural map B). Using Theorem 7.2.1.5 we can canonically identify the fiber at f : A ! B with

Finally, we observe that there is a canonical map

S 1 -algebras

We now introduce the second major character of the HKR equivalence, namely the 1-category of S 1 -algebras.

As for mixed algebras, we have at our disposal two different description for this 1-category, one that is naturally monadic and another one which is naturally comonadic. Again, as for mixed algebras, the one we are truly interested in is the monadic one, because it encodes Hochschild homology. However, the comonadic one is easier to study and manipulate. For this reason, we devote this section to the study of the equivalence between the two presentations.

We start with some general considerations. We consider Z as a discrete, grouplike E 1 -monoid in S. The 1-categorical version of May's theorem (see [Lur12c,5.2.6.15]) provides an equivalence Bar (1) :

where S ⇤ is the 1-category of pointed spaces and S We denote by U : Mon gp E1 (S ⇤ ) ! S the forgetful functor and we set B := U Bar (1) .

We therefore define

Notice that the underlying homotopy type of S 1 coincides with the one coming from its standard model, i.e. we have an equivalence S 1 ' ⇤ q ⇤q⇤ ⇤ in S. Defining S 1 := B(Z) has merely the advantage of explicitly fixing the E 1 -structure on S 1 . Observe now that since S 1 is again a group-like E 1 -monoid, we are entitled to consider B(S 1

), which inherits the same kind of structure. When needed, we will therefore consider B(S 1

) as a group-like E 1 -monoid. Notice that there is a canonical map

), corresponding to the unit in B(S 1

). Precomposition with u provides us with a forgetful functor

Suppose that C is complete and cocomplete. Then the functor u S 1 : S 1 -C ! C is conservative and admits both a left and a right adjoint. In particular, u S 1 is both monadic and comonadic.

Proof. The left (resp. right) adjoint to u S 1 is given by left (resp. right) Kan extension along u : ⇤ ! B(S 1 ). We are therefore left to check that u S 1 is conservative. This follows at once because u is an effective epimorphism.

Our next goal is to identify the monad and the comonad associated to u S 1 : S 1 -C ! C. Observe that the situation is dual: switching from C to Cop interchanges the monad and the comonad. It is therefore enough to focus on the description of the monad.

A rectification result for S 1 -object in spaces

We start our investigation in the simplest case possible, namely when C coincides with the 1-category of spaces S. In this situation, S 1 is an internal group object in C. We are therefore allowed to form the 1-category LMod S 1 (S). This category is equipped with a forgetful functor v S 1 : LMod S 1 (S) ! S, which is obviously monadic. Both the categories S 1 -S and LMod S 1 (S) encode the idea of spaces with the action of S 1 . It is therefore reasonable to expect them to be equivalent. Remark 7.2.2.3. Let us denote by T S 1 (resp. R S 1 ) the monad associated to u S 1 (resp. v S 1 ). It is easy to verify that these two monads have the same underlying endofunctor. To see this, start by observing that B(S 1

) /⇤ ' ⇤ ⇥ B(S 1 ) ⇤ ' S 1 . Therefore, the formula for the left Kan extension yields for every X 2 S:

On the other hand, ?? implies that R S 1 (X) ' S 1 ⇥ X. As S is an 1-topos, we have a canonical equivalence

This implies that the two endofunctor are equivalent.

In virtue of the above remark, all we have to do is to verify that the equivalence between the endofunctors can be lifted to an equivalence between the monads. This is however less obvious than one might expect. Our proof passes through the simplicial description of the 1-category LMod S

spanned by the functors F : op ⇥ 1 ! S for which the arrows

Consider now the trivial Cartesian fibration

The functor G determines a map g : op ! S ⇥ op compatible with the projection over op. We let E be the 1-category fitting in the following pullback diagram:

We claim that q : E ! op is a Cartesian fibration. Indeed, [Lur09c,2.4.7.12] shows that ev 1 : Fun( 1, S) ! S is a Cartesian fibration. Therefore, the stability of Cartesian fibrations under base change (see [Lur09c,2.4.2.3]) implies first that ev 1 ⇥ id op is a Cartesian fibration, and then that the same goes for q.

Inspection reveals that the fiber of q at [n] 2 op is equivalent to the 1-category S /(S 1 ) ⇥n . Unraveling the definitions, we see that a morphism

) ⇥m in E is q-Cartesian if and only if it is a pullback square and the morphism (S 1 ) ⇥n ! (S 1 ) ⇥m is equivalent to G(s) for some s : [n] ! [m] in op. In turn, this implies that the associated unstraightened functor

Un(q) :

! Cat 1 can be informally described as the functor It follows from the description we gave above of the q-Cartesian edges that we can identify LMod S 1 (S) with the full subcategory of the left hand side spanned by Cartesian sections. Therefore, [Lur09c,3.3.3.2] provides us with the following chain of equivalences:

On the other hand, we observe that the inclusion S ✓ Cat 1 has both a left and a right adjoint. In particular, it commutes with colimits. As a consequence,

), S) ' lim Fun(G, S).

Using [Lur09c, 2.2.1.2], we see that for every [n] 2 , we have a natural identification

) ⇥n , S) ' S /(S 1 ) ⇥n .

In other words, we can identify Fun(G, S) : op ! Cat 1 with the functor Un(q) introduced above. This completes the proof. (ii) a spectrum object E 2 Sp(Geom(C, ⌧, P)).

Furthermore, we impose the following two conditions on E:

(i) the canonical map ⌦ 1 (E) ! ⇤ is in P;

(ii) for every n > 0, the canonical morphisms ⇤ ! ⌦ 1 n (E) is a P-atlas;

We say that a pre-T-nonconnective context is a T-nonconnective context if the following additional condition is satisfied:

(3) for every X 2 C and every n , one has ⇡ 0 Map Geom(C,⌧,P) (X, ⌦ 1 n (E)) ' ⇤.

Notation 7.3.1.7. Let (T, adm, ⌧) be a pregeometry and let (C, P, E) be a T-nonconnective context. We set

In particular, we have the relation

). (i) when T = T ét (k), we take E alg := {B n (G a )}, the spectrum associated via May's delooping theorem to the commutative k-group scheme G a . As the relations

are satisfied, we see that {B n (G a,k )} n 0 form indeed a spectrum object. Furthermore, the morphisms Spec(k) ! B n (G a,k ) are smooth atlases.

(ii) When T = T an (C) or T = T an (k), we take

Here G a denotes the analytic affine line A 1 k ' (A 1 k ) an , seen as an analytic commutative group. As in the algebraic setting, this is indeed a spectrum object, and each B n (G a ) is a geometric stack with smooth atlas given by Spec(k) ! B n (G a ).

(iii) When T = T an (k) for a nonarchimedean field k equipped with a non-trivial valuation we have many natural choices for E. Indeed, for any r 2 R >0 , we can consider the disk D 1 k (0, r). Since the k is nonarchimedean, we see that D 1 k (0, r) is an abelian group object in T an (k) and therefore we can consider its delooping stacks B n (D 1 k (0, r)). We denote the spectrum {B n (D 1 k (0, r))} n 0 by E(r), with the understanding that when r = 1 we get back the spectrum of the previous example. We will see later that the extra freedom in choosing the spectrum E is one of the keys to the nonarchimedean HKR in many interesting situations (such as when k = Q p ). Whenever a T-nonconnective context is fixed, we can define T-nonconnective structures. Notation 7.3.1.9. Let (T, adm, ⌧) be a pregeometry and let (C, P, E) be a pre-T-nonconnective context. We let T nc be the smallest full subcategory of Geom(C, ⌧, P) closed under finite products and containing the objects of the form Spec T (X) and the geometric stacks E n for n 0. (ii) O preserves the pullbacks of the form

We denote the 1-category of T nc -structures on X and local transformations between them by Str loc T nc (X). Remark 7.3.1.12. In what follows we will mainly restrict our attention to nonconnective structures for a nonconnective pregeometry T nc . In other words, we almost always work with T-nonconnective contexts rather than pre-T-nonconnective contexts. There is only one exception: in Section 7.3.8 it is important to allow pre-Tnonconnective contexts. This is the reason we formulated the above definition in this more general setting.

The above definition is justified by Theorem 7.3.4.1, that proves in particular that if k is a field of characteristic zero, then the 1-category of T nc ét (k)-structures on X coincides with the 1-category of sheaves with values in the 1-category of cdgas.

However, before stating and proving this result, we need to study some general features of the 1-category of nonconnective structures.

Underlying spectrum object

Let (T, adm, ⌧) be a pregeometry and let (C, P, E) be a T-nonconnective context. Using [Lur09c, 5.3.6.2] we find an 1-category G nc equipped with an 1-functor j : T nc ! G nc enjoying the following properties:

(i) the 1-category G nc is idempotent complete and admits finite limits;

(ii) the functor j commutes with products, admissible pullbacks in T and take diagrams of the form (7.3.1.1) to pullbacks;

(iii) for every other idempotent complete 1-category with finite limits E, composition with j induces an equivalence

Fun lex (G, E) ! Fun 0 (T nc , E), where Fun 0 (T nc , E) denotes the full subcategory of Fun(T nc , E) spanned by those functors commuting with products, admissible pullbacks in T and taking diagrams of the form (7.3.1.1) to pullbacks;

(iv) j is fully faithful.

Since the inclusion T ✓ Geom(C, ⌧, P) commutes with products, admissible pullbacks and takes the diagrams (7.3.1.1) to pullbacks, we obtain a canonical left exact functor i : G nc ! Geom(C, ⌧, P).

Let p : G ! Nop be the Cartesian fibration associated to the diagram

and let q : C ! Nop be the Cartesian fibration associated to

Since i : G nc ! Geom(C, ⌧, P) is left exact, it induces a morphism of Cartesian fibrations f : G ! C. Let us represent an object in G (resp. in C) by a pair (X, n) where n 2 N and X 2 G nc (resp. X 2 Geom(T, ⌧, P)). We However, as the previous example shows, there is no natural T-structure on the spaces ⌦ 1 n (A). Furthermore, the additional structure coming from E might consist of significantly less operations than the ones provided by T. For example, we will see later in the paper that a T an (k)-structure essentially consists of an unbounded cdga A equipped with an analytic structure on ⌧ 0 (A). Proposition 7.3.2.5. The forgetful functor U : Str T nc (X) ! Sp(X) commutes with limits and filtered colimits. Furthermore, suppose that Str T (X) is closed under sifted colimits in Fun(T, X). Then U commutes with sifted colimits.

Proof. Let us first observe that since limits and filtered colimits commute with finite limits, the inclusion Str T nc (X) ,! Fun(T nc , X) preserves limits and filtered colimits. Since limits and colimits in a category of functors are computed objectwise and U is given by evaluation, the statement follows immediately.

Suppose now that Str T (X) is closed under sifted colimits in Fun(T, X). It is enough to prove that if

) is again a T nc -structure. Observe that O commutes with finite products because sifted colimits commute with finite products. Furthermore, the assumption implies that the restriction of O to T is a T-structure. We are therefore left to check that

)).

Notice that using the 1-categorical version of May's delooping theorem we can factor evaluation at E n through

Mon gp E n (X). In this way, the looping functor ⌦ gets identified with the forgetful functor

As this forgetful functor commutes with sifted colimits, the conclusion follows.

Connective covers

Let (T, adm, ⌧) be a pregeometry and let (C, P, E) be a T-nonconnective context. Let X be an 1-topos. Precomposition with the natural inclusion j : T ,! T nc induces a well defined functor

We refer to this functor as the connective cover functor.

Since both Str T (X) and Str T nc (X) are presentable and ⌧ 0 commutes with limits and filtered colimits, the adjoint functor theorem implies the existence of a left adjoint i to ⌧ 0 . The goal of this section is to study the properties of the functor i. In particular, we will prove that in many cases i is fully faithful, and we will provide a characterization of its essential image.

We start by providing a sufficient criterion to check that a functor O : T nc ! X is a T nc -structure. Let us begin by fixing some notation. Let D be a Cartesian symmetric monoidal 1-category. Using [Lur12c,4.1.2.11] we can identify the 1-category Mon E1 (D) of E 1 -monoid objects in D with the full subcategory of Fun( op, D) spanned by those simplicial objects satisfying the Segal condition (cf. [Lur12c, 4.1.2.5]). We denote by B the corresponding functor:

When D admits geometric realization of simplicial objects, then we have a natural transformation of functors

( ) that is in fact an equivalence.

Suppose now given a product-preserving functor O : T nc ! X. As E is a spectrum object in Geom(C, ⌧, P), we see that each E n acquires the structure of a grouplike (E 1 and hence) E 1 -monoid in T nc . Since O respects the Cartesian structures, we see that the canonical morphism in Fun( op, X ⇤ )

is an equivalence. On the other hand, we can identify B(E n ) with the Čech nerve of the P-atlas ⇤ ! E n+1 . We therefore obtain a canonical morphism

).

With these notations, we can now prove the following result:

, X) be a functor. Suppose that:

(i) the restriction O| T is a (local) T-structure;

(ii) the functor O commutes with products;

(iii) for every n 1, the canonical morphism

) is an equivalence X;

Then O is a (local) T nc -structure.

Proof. We only need to check that the canonical morphism

is an equivalence for every n 0. Since O commutes with products, the canonical morphism

is an equivalence. Coupling this observation with the hypothesis on n O , we deduce that the canonical morphism

In virtue of May's delooping theorem [Lur12c, 5.2.6.15], we see that the canonical morphism

)) is an equivalence in Mon gp E1 (X ⇤ ). Applying the forgetful functor to X ⇤ and [Lur12c, 5.2.6.12], we finally obtain that the morphism (7.3.3.1) is an equivalence, thus completing the proof.

Later in this section we will characterize T-structures exactly as those T nc -structures satisfying condition (3) in the above lemmama. The following example illustrates why it is reasonable to expect a similar characterization:

Example 7.3.3.2. Let X 2 T be any object. Then the functor A nc X : T nc ! S given by A X (Y ) := Map T nc (X, Y ) is a T nc -structure. Furthermore, assumption (3) in Theorem 7.3.1.6 implies that for every n 1 one has:

In other words, the morphism

) is an effective epimorphism. Observe now that A nc X commutes with all limits. In particular, it commutes with the Čech nerve Č(p) of the P-atlas p : ⇤ ! E n . This implies that we can identify the simplicial object

As we already argued that this is an effective epimorphism, we finally conclude that n A nc X is an equivalence. Combining Theorem 7.3.3.1 and the above example we can produce many T nc -structures out of T-structures. Before stating the result, let us introduce some notations. We let

denote the canonical inclusions and we denote by L X and L nc X their left adjoints, respectively. When X = S is the 1-topos of spaces, we omit the X in the subscript.

Proof. To simplify the notations, we simply write Lan j (O) instead of Lan j (◆ O). Notice that the functor ◆ : Str T (X) ! Fun(T, X) commutes with sifted colimits. As Lan j : Fun(T, S) ! Fun(T nc , S) is a left adjoint, we conclude that

As O ↵ ' A X↵ , we have

In particular, each Lan j (O ↵ ) is a T nc -structure. As I is sifted, we easily conclude that Lan j (O) commutes with finite products. Furthermore, for each ↵ 2 I, Theorem 7.3.3.2 implies that the natural morphism

is an equivalence for every n 1. Since I is sifted, we see that

Since colimits commute with colimits, we conclude that n Lanj (O) is an equivalence as well. Therefore, Theorem 7.3.3.1 implies that Lan j (O) is a T nc -structure.

is an equivalence for every n 1. As in an 1-topos every groupoid object is effective, we conclude that B(O(E n )) can be identified with the Čech nerve of the map

). In other words, we have

Observe now that the adjunction (i, ⌧ 0 ) provides a counit map " O : i(⌧ 0 O) ! O. As i is fully faithful, we see that this map induces an equivalence on the connective covers. As both functors commute with products, we see that it is enough to check that " O is an equivalence when evaluated on E n for every n 1. This is easily checked by induction on n, starting with n = 0 and using the fact that both O and i(⌧ 0 O) commute with the geometric realization of the Čech nerves of the maps p n+1 : ⇤ ! E n+1 .

Corollary 7.3.3.5. Suppose that the pregeometry T is discrete. Then for every 1-topos X the functor i : Str T (X) ! Str T nc (X) is fully faithful.

Proof. It is enough to prove that the composition

Suppose first that X = PSh(D) is the 1-topos of presheaves on an 1-category D. For every d 2 D let commutes. Indeed, this is because the functors ev d commute with arbitrary colimits (in particular, the ones computing the left Kan extension along j). As we saw in the proof of Theorem 7.3.3.4, the bottom row factors through Str T nc (S).

We now deal with the general case. Choose a presentation of X as left exact accessible localization of an 1-category of presheaves, y : X PSh(D) : . Let us write Lan X j and Lan D j to denote the following left Kan extension functors Lan X j : Fun(T, X) ! Fun(T nc , X) , Lan D j : Fun(T, PSh(D)) ! Fun(T nc , PSh(D)).

Similarly, let us write ⇤ and nc ⇤ to denote the following functors, given by composition with :

With these notations, the diagram Fun(T, PSh(D)) Fun(T nc , PSh(D))

Fun(T, X) Fun(T nc , X)

Lan X j commutes. Let y ⇤ : Fun(T, X) ! Fun(T, PSh(D)) denote the functor given by composition with y. Then the commutativity of the previous diagram and the full faithfulness of y ⇤ imply that

As the functor nc ⇤ commute with finite limits, it preserves T nc -structures. It is therefore sufficient to prove that the composition

Str T (X) Fun(T, X)

commutes, we are reduced to the case where X = PSh(D), that has already been dealt with previously.

Proof. The functor ⌧ 0 : Str T nc (X) ! Str T (X) commutes with limits and filtered colimits, and both categories are presentable. It follows that ⌧ 0 admits a left adjoint, that we denote i. Recall that [Lur11d, Proposition 3.3.1] proves that Str loc T (X) is closed under sifted colimits in Fun(T, X), and hence in Str T (X). Therefore, the same argument of Theorem 7.3.3.6 implies that the composition

i is fully faithful. In particular, it factors through Str loc T nc (X) and the resulting functor coincides with the left adjoint to ⌧ 0 : Str loc T nc (X) ! Str loc T (X) we constructed in Theorem 7.3.3.6. The conclusion follows. Proposition 7.3.3.8. For any 1-topos X, the forgetful functor

is conservative as well.

Proof. Let ' : O ! O 0 be a morphism of nonconnective T-structures. If U (') is an equivalence, then unraveling the definition of the underlying spectrum we deduce that

is an equivalence. On the other hand, whenever X 2 T, Corollaries 7.3.3.5 and 7.3.3.6 imply that

Therefore ' X : O(X) ! O 0 (X) is an equivalence as well. The conclusion now follows from the fact that both O and O 0 commute with finite products.

Nonconnective structures in the algebraic case

In this section we focus on the special case where T = T disc (k) and the nonconnective context is the one of Theorem 7.3.1.8 [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. We can summarize the main results as follow:

(i) when k is a ring containing Q, we provide a canonical equivalence between the 1-category of nonconnective T disc (k)-structures and the 1-category of cdga's;

(ii) when k is a field of positive characteristic, we provide a fully faithful embedding of cohomologically connected cosimplicial algebras in the 1-category of nonconnective T ét (k)-structures.

Let us start by assuming that k contains Q. In this case there is a canonical model structure on cdga k . We denote the underlying 1-category by cdga k . We know that

and this equivalence is realized by sending a T disc (k)-structure to its evaluation on A 1 k 2 T disc (k). In particular, we see that the inclusion Str T disc (k) (S) ,! Fun(T disc (k), S) commutes with sifted colimits. As a consequence, we can invoke Theorem 7.3.2.5 to deduce that the underlying spectrum U : Str T disc (k) nc (S) ! Sp commutes with sifted colimits. Furthermore, since the equivalence above is realized by evaluation on A 1

k ' E 0 , we deduce from Theorem 7.3.3.8 that the underlying spectrum functor U is also conservative. It follows that U is monadic.

Notice that Corollaries 7.3.3.6 and 7.3.3.7 imply that the composition

is fully faithful. Therefore, the chain rule for Goodwillie derivatives implies that the induced functor

is fully faithful. Pairing this with the equivalence (7.3.5.1), we deduce that the functor

is fully faithful. We summarize this in the following:

Proposition 7.3.5.1. Let X be an 1-topos with enough points and let A 2 Str loc T (X) be a connective local T-structure. Then the functor

is fully faithful. Furthermore, it has a right adjoint given by @(⌧ 0 ).

Proof. The fact that (@(i), @(⌧ 0 ) can be promoted to an adjoint pair is a standard consequence of Goodwillie's calculus. The full faithfulness of @(i) follows from the above discussion.

Remark 7.3.5.2. It seems likely that the above adjunction is actually an equivalence. However, for our purposes, full faithfulness is largely sufficient.

We now use Theorem 7.3.5.1 to develop the theory of nonconnective cotangent complex. Definition 7.3.5.3. Let X be an 1-topos and let A 2 Str T nc (X). We refer to the functor

as the T-theoretic nonconnective split square-zero extension functor (or simply as the nonconnective split squarezero extension functor when T is clear from the context). Given M 2 Sp(Str T nc (X) /A ) we set

When T nc is clear from the context, we simply write A M to denote this nonconnective T-structure. Notice that the the functor Der T nc (A; ) commutes with limits and with -filtered colimits for  a sufficiently big regular cardinal. The adjoint functor theorem implies therefore that it is corepresentable. Definition 7.3.5.5. Let X be an 1-topos and let A 2 Str T nc (X). We denote the corepresentative of the functor Der T nc (A; ) by L T nc A and we refer to it as the T nc -theoretic cotangent complex (or as the nonconnective Ttheoretic cotangent complex).

When A is a T-structure we therefore have at our disposal two cotangent complexes, L T A (cf. [PY17b, Definition 5.4]) and L T nc A . Our next goal is to prove that these two objects are canonically equivalent.

Lemma 7.3.5.6. Let X be an 1-topos and let A be a T nc -structure on X. The connective cover functor ⌧ 0 : Str T nc (X) !

In other words,

Proof. This simply follows from the fact that ⌧ 0 , being a right adjoint, commutes with limits.

Proposition 7.3.5.7. Let X be an 1-topos with enough points. Let A 2 Str loc T (X). Then there is a canonical equivalence

in the 1-category Sp(Str T nc (X) /i(A) ). In particular, L T nc i(A) belongs to the full subcategory Sp(Str T (X) /A ).

Proof. Let M 2 Sp(Str T nc (X) /i(A) ). Then, using Theorem 7.3.5.6, we obtain

We can therefore deduce from the Yoneda lemmama that @(i)(L T A ) ' L T nc A . The second statement follows from the full faithfulness of @(i).

Change of spectrum

The leitmotiv of this paper is to provide an axiomatic context where to formulate and prove the HKR theorem. In later sections we will provide four different contexts where our formalism applies. Given the abundance of such HKR theorems, a very natural question is to compare them whenever the question makes sense. In order to carry out such a task, we need to introduce a suitable notion of transformation of nonconnective contexts. Furthermore, the ideas introduced in this context will prove fundamental to prove certain cases of the analytic HKR (namely, when the residue field has positive characteristic).

We start by analyzing a simple situation, where we keep the pregeometry T and the T-geometric context C fixed, but we change the spectrum E. Consider therefore two spectra E 0 , E 1 2 Sp(Geom(C, ⌧, P)) satisfying the assumptions of Theorem 7.3.1.6. We denote by T nc i the nonconnective pregeometry generated by E i for i = 0, 1. Define T nc 01 to be the smallest full subcategory of Geom(C, ⌧, P) closed under products and containing E n i for i = 0, 1 and n 0. Notice that we have fully faithful embeddings j i : T nc i ! T nc 01 for i = 0, 1. We define a T nc 01 -structure in an 1-topos X to be a functor

which commutes with products and whose restrictions to T nc i are T nc i -structures in the sense of Theorem 7.3.1.11.

Proof. The first statement follows immediately from Theorem 7.3.6.2 by passing to left adjoints. The second statement follows from the fact that under the assumption on T nc i ,! T nc 01 we have a commutative diagram

Fun(T nc i , X) Fun(T nc 01 , X).

i

Lan

In other words, we can identify the functor i with the left Kan extension. As a consequence, we have

because the left Kan extension is fully faithful.

Corollary 7.3.6.5. Suppose that left Kan extension along T nc 0 ,! T nc 01 takes T nc 0 -structures to T nc 01 -structures. Then the diagram

Proof. As 01 = ⇢ 1 0 , this is a direct consequence of Propositions 7.3.6.2 and 7.3.6.3. In virtue of the above results, it is useful to have a more geometrical condition only involving the spectra E 0 and E 1 implying that left Kan extension along T nc i ,! T nc 01 takes T nc i -structures to T nc 01 -structures. Definition 7.3.6.6. Let (T, adm, ⌧) be a pregeometry and let (C, ⌧, P) be a T-geometric context. We say that (C, ⌧, P) is ordinary if the topology ⌧ is quasi-compact and there exists a full subcategory C 0 ✓ C satisfying the following conditions: (i) C 0 is a 1-category with finite limits;

(ii) the topology ⌧ and the class of morphisms P restrict to C 0 and make (C 0 , ⌧, P) into a geometric context;

(iii) T is contained in C 0 .

We say that a nonconnective pregeometry T nc is ordinary if the underlying T-geometric context is ordinary.

Example 7.3.6.7. All the contexts introduced so far are ordinary. The choice of C 0 is in each case canonical and it corresponds to the full subcategory spanned by discrete objects. For instance, when T = T ét (k), C 0 is the category of classical affine schemes. When T = T an (k), C 0 is the category of classical k-analytic spaces.

In virtue of the above example, whenever (C, ⌧, P) is an ordinary T-geometric context, we refer to objects in C 0 as the discrete objects. Furthermore, we refer to the stacks in Geom(C 0 , ⌧, P) as the geometric stacks.

When the T-nonconnective context is ordinary we have an extra amount of control on the spectrum E, as the following couple of result shows. be the colimit of F computed in PSh(C 0 ). Suppose that there exists an n 0 such that for each ↵ 2 I, the geometric stack F ↵ := F (↵) is n-truncated. Then F 1 satisfies ⌧ -hyperdescent.

Change of context

We now turn to the general case, where we do not keep underlying pregeometry fixed. Definition 7.3.7.1. Let T nc 0 and T nc 1 be two nonconnective pregeometries. A transformation of nonconnective pregeometries from T nc 0 to T nc 1 is the given of a morphism of geometric contexts ' : (C 0 , ⌧ 0 , P 0 ) ! (C 1 , ⌧ 1 , P 1 ) satisfying the following two conditions: (i) the morphism of geometric contexts ' restricts to a transformation of pregeometries T 0 ! T 1 ;

(ii) let T 2 be the T 1 -nonconnective context determined by (C 1 , ⌧ 1 , P 1 ) and the spectrum object ' s (E 0 ). Then the left Kan extension along T nc 2 ,! T nc 12 takes T nc 2 -structures to T nc 12 -structures. We denote a transformation of pregeometries by ' : (C 0 , P 0 , E 0 ) ! (C 1 , P 1 , E 1 ). Definition 7.3.7.2. A transformation of nonconnective pregeometries ' : (C 0 , P 0 , E 0 ) ! (C 1 , P 1 , E 1 ) is said to be a change of spectrum if the underlying transformation of geometric contexts is an equivalence. Definition 7.3.7.3. A transformation of nonconnective pregeometries ' : (C 0 , P 0 , E 0 ) ! (C 1 , P 1 , E 1 ) is said to be strong if there is an equivalence

Notice that if ' is a strong transformation of nonconnective pregeometries it induces a well defined functor

In this paper we abusively identify the strong transformation with the functor ' : T nc 0 ! T nc 1 . It follows from the definitions that we can always factor a transformation of nonconnective pregeometries as a strong transformation followed by a change of spectrum. We already performed an in-depth analysis of the change of spectrum situation in the previous section. Therefore, we focus now on strong transformations.

Let ' : T nc 0 ! T nc 1 be a strong transformation of pregeometries. For any 1-topos X, precomposition with ' induces a well defined functor ' ⇤ : Str T nc 1 (X) ! Str T nc 0 (X). This functor commutes with limits and filtered colimits. In particular, it admits a left adjoint that we denote ' ⇤ . Proposition 7.3.7.4. Let ' : T nc 0 ! T nc 1 be a strong transformation of pregeometries. For any 1-topos X and any

Proof. This is an immediate consequence of the fact that ' ⇤ commutes with (not necessarily) finite limits.

Corollary 7.3.7.5. Let ' : T nc 0 ! T nc 1 be a strong transformation of pregeometries. For any 1-topos X, any

Here @(' ⇤ ) denotes the Goodwillie derivative of ' ⇤ .

Proof. This follows immediately from the fact that @(' ⇤ ) is left adjoint to @(' ⇤ ) and from the commutativity of the diagram asserted in the previous proposition. Indeed, the canonical map is simply the Beck-Chevalley transformation.

Morita equivalences

Fix a pregeometry (T, adm, ⌧) and an 1-topos X. We are often more interested in manipulating the 1-category

Str loc

T (X) than the 1-category Str T (X). The reason T-structures are only a tool needed to set up the theory of T-structured spaces and, ultimately of T-schemes.

Unfortunately, the 1-category Str loc T (X) is typically not presentable. This can be fixed as follows: for any A 2 Str loc T (X), the 1-category Str loc T (X) /A becomes presentable (see [Por15b,Corollary 9.4]). This fact has been used in an extensive way to obtain several of the main results of [PY17b].

On the other hand, Str loc T (X) is much more flexible of Str T (X). Indeed there are typically several different pregeometries giving rise to the same 1-categories of local structures. This has been studied in [Lur11d,§3.2].

The same picture applies when dealing with nonconnective pregeometries. In the nonconnective setting the Morita equivalence plays an even greater role. To understand the reason consider the following example: (BG a )} n 0 defines a spectrum object E in Geom(C, ⌧ ét , P sm ), and (C, P sm , E) becomes a pre-T an (k)-nonconnective context. Nevertheless, this is not a T an (k)-nonconnective context, because it is not true that for every X 2 T an (k) one has H 1 (O X ) = 0 (take for example X = P 1 k ). In this situation, ?? is no longer valid. 3 In order to fix this result one can change the starting pregeometry T an (k) as follows: we define T an (k) 0 to be the category of smooth k-affinoid spaces. The same choices of admissible morphisms and of the topology for T an (k) endow T an (k) 0 with a the structure of a pregeometry. Furthermore, the inclusion T an (k) 0 ! T an (k) is a Morita equivalence, and (dAn k , ⌧ ét , P sm ) is a T an (k) 0 -geometric context. Therefore, (dAn k , P sm , E) is a pre-T an (k) 0 -nonconnective context. The difference with before is that now Tate's acyclicity theorem implies that

The conclusion is that although the results we developed in the previous sections might not apply to T an (k)structures, they will apply for T an (k) 0 -structures, and thanks to Morita equivalence, they will also apply to T an (k)local structures. 4 Motivated by the previous example, we introduce the notion of Morita equivalence. Definition 7.3.8.2. Let (T 0 , adm 0 , ⌧ 0 ) and (T 1 , adm 1 , ⌧ 1 ) be two pregeometries. For i = 0, 1, let (C i , P i , E i ) be a pre-T i -nonconnective context. A Morita equivalence of pre-nonconnective contexts from (C 0 , P 0 , E 0 ) to (C 1 , P 1 , E 1 ) is the given of a Morita equivalence of geometric contexts

with the following properties:

(i) it restricts to a Morita equivalence of pregeometries ' : T 0 ! T 1 ;

(ii) under the equivalence Geom(C 0 , ⌧ 0 , P 0 ) ' Geom(C 1 , ⌧ 1 , P 1 ) the spectrum E 0 is equivalent to E 1 .

Let ' : (C 0 , P 0 , E 0 ) ! (C 1 , P 1 , E 1 ) be a transformation of pre-nonconnective contexts. Then ' induces a functor

Precomposition with this functor provides, for every 1-topos X, restriction morphisms

The following is the analogue of [Lur11d, Proposition 3.28] in the nonconnective setting:

In the rigid analytic situation, there are other variations that are important for us. For every r 2 |k ⇥ | let D 1 k (r) denote the closed disk of radius r centered at the origin. Since the valuation is non-archimedean, D 1 k (r) acquires the structure of a commutative group object in dAn k . Applying once again May's delooping theorem for 1-topoi, we obtain a spectrum object

). Similarly to the case discussed above, (dAn k , ⌧ ét , P sm , E(r)) is then a pre-T an (k)-nonconnective context and also similary (dAfd k , ⌧ ét , P sm , E(r)) is a pre-T an (k)-nonconnective context, moreover we have the following: Lemma 7.4.1.3. The pre-T an (k)-nonconenctive context (dAfd k , ⌧ ét , P sm , E(r)) is a T afd an (k)-nonconnective context.

Proof. We are left to check condition (3) in Theorem 7.3.1.6. Let X 2 dAfd k be a derived k-affinoid space, we want to show that the mapping space

is connected, whenever n 1 and r 2]0, 1] is a given radius. Let us treat first the case where r = 1, in this case E n (r) ' B n (BG a ). By the universal property of the k-analytic group BG a together with the universal property of the delooping functor we obtain an equivalence between spaces

. By definition of derived k-affinoid space we know that, for each j 0, ⇡ j (O X ) is a coherent sheaf over the 0-truncated k-affinoid space t 0 X = (X, ⌧ 0 O X ). Moreover, thanks to (the dual of) [Lur12c, 1.2.2.14] we have a spectral sequence of the form

) and by Tate's acyclicity theorem it follows that we have an equivalence of spaces

whenever n > 0. Suppose now that r < 1, by May's delooping theorem we reduce ourselves to show that the mapping space

is a discrete space, i.e., the only non-trivial homotopy groups live in degree 0. We have a monomorphism D 1 k (r) ! BG a in the 1-category dAfd k and therefore for each n 1 we have monomorphisms B n (D 1 k (r)) ! B n BG a and therefore we have a monomorphism of mapping spaces

2) and we can identify the left hand side of (7.4.1.2) with global sections of the subsheaf ⇡ 0 (O X ) (r) of ⇡ 0 (O X ) on t 0 X spanned by those sections which (locally) are uniformly bounded in norm by r > 0.Therefore, covering t 0 X by a finite number of open affinoid spaces X = S i X i , with (classical) intersections X i,j = X i \ X j we have a commutative diagram of the form

where the vertical maps are monomorphisms and A i,j (r

commutes with sifted colimits. Using [Lur11b, Proposition 2.11], we reduce ourselves to the case where A is an elementary T an (k)-structure. In this case, the formula follows from direct inspection.

Let us fix A 2 Str loc Tan(k) (X). Since ⌧ 0 (i(A)) ' A and ⌧ 0 (i(A alg )) ' A alg , we obtain the following commutative square:

Using [Por15b, Corollary 9.4] we see that all the categories appearing in this diagram are presentable. As the functors ( ) alg commute with limits and filtered colimits, we deduce that they admit left adjoints, that we still denote by

T nc an (k) (X) /i(A) . We refer to the functor on the right as the nonconnective (derived) analytification functor.

These functors induce the following commutative diagram:

In particular, we see that the nonconnective analytification functor respects connective structures.

Remark 7.4.2.3. The functor ( ) an : Str loc

commutes. Here we denoted by A ( ) an and B ( ) an the analytification functors constructed using A (resp. B) as auxiliary choice, and f ⇤ and f alg ⇤ denote the functors induced by composition with f (resp. f alg ⇤ ). For this reason we suppress the dependence on A in the notation for the analytification functor.

If r  ⇢, where ⇢ is the converging radius of the exponential, then the Beck-Chevalley transformation

) is an equivalence in Str T nc an (k) (X). Proof. By definition, we have

) ' S an k (m, r), and therefore

, we see that we have to prove that the canonical morphism

is an equivalence in Str T nc an (k) (X) In order to check this, we can reason locally on X. In particular, we can assume X to be a derived analytic space. Let us write X = (X, O X ), so that Str T nc an (k) (X) ' Str T nc an (k) (X). As O X 2 Str loc Tan(k) (X), we can use [Lur11b,Proposition 2.11] to further reduce ourselves to the case where X 2 T an (k). At this point, the conclusion follows from the Yoneda lemma and the fact that T nc an (k)-structures commute with products by assumption.

Let X = (X, O X ) be a derived analytic space. Let q ⇤ : X ⌧ S : q 1 be the canonical geometric morphism. We set S an X (m, r) := q 1 (S an k (m, r)).

As observed in the previous proof, we have a canonical identification q ⇤ p ⇤ (O B(D m k (r)) ) ' O X b ⌦ k S an X (m, r). We first address the question of whether this is a local structure or not. ). We observe now that we can explicitly represent ⌧ 0 (q 0 ⇤ (O X⇥B(D m k (r)) )) as the functor T an (k) ! X sending Y 2 T an (k) to the sheaf on X defined by sending an étale morphism U ! X from a derived Stein (resp. kaffinoid) U to We claim that this square is a pullback. Observe that Map dAnSt k (U ⇥ B(D m k (r)), V ) ' lim We just argued that the back square is a pullback. A similar (but easier) use of [Lur11d,Remark 2.3.4] shows that the front square is also a pullback. Therefore, in order to prove that the top square is a pullback, it is enough to prove that the bottom square is a pullback. We now observe that both Y and t 0 (U ) ⇥ B(D m k (r)) are underived. Furthermore, the functor of points of Y takes values in sets. are equivalences, and hence that this diagram is a pullback. This completes the proof.

We now use the results of ?? to extend the above result to a more general situation. Using ??, we can write

where E ⇢ Str loc T (S) /x 1 O is the full subcategory spanned by germs of T an (k)-structures (see ??). As a consequence, we obtain

As O 0 is a germ of a T-structure, we can apply Theorem 7.4.3.2 to deduce that ⌧ 0 (O 0 b ⌦ k S an k (m, r)) is a local structure. Using ?? again, we see that the 1-category E is filtered. As ⌧ 0 commutes with filtered colimits, we finally conclude that

As local structures are closed under filtered colimits, the conclusion follows.

Relative Van Est

Let X be an 1-topos with enough points and let O 2 Str loc Tan(k) (X) be a local T an (k)-structure. Let p ⇤ : X S : p 1 be the canonical geometric morphism. We set S an X (n, r) := p 1 S an k (n, r).

Recall that (S an k (n, r)) alg ' S k (n). There is a canonical morphism

The main result of this section is the following: We prove this theorem by several reduction steps. Consider the full subcategory C X of Str loc Tan(k) (X) spanned by those O 2 Str loc Tan(k) (X) such that ⌘ O,n,r is an equivalence. Notice that O 2 C X if and only if for every geometric point x ⇤ : S X : x 1 , one has x 2 O 2 C S . We can therefore suppose that X = S. Recall now that both the functor b ⌦ k S k (n, r) and ( ) alg commute with filtered colimits. Using ??, we are therefore reduced to prove the theorem in the case where O ' x 1 O X for X 2 dAn k and x ⇤ : S X : x 1 a given geometric point. For this, it is enough to prove that

is an equivalence in Str loc T nc an (k) (X X ). We proceed by induction on the Postnikov tower of X. When X is underived, the result follows from Theorem 7.4.2.9. Suppose therefore that the statement has been proven for X n := t n (X).

We have a pullback diagram in Str loc

Tan(k) (X)

where M := ⇡ n+1 (O X )[n + 2] and where d 0 corresponds to the zero derivation. As both functors ( ) alg and ⌦ k S k (n) commute with limits, we conclude that the diagram

is again a pullback.

We now claim that the diagram ⌦ k S an k (n, r). We now observe that in the 1-category of derived analytic geometric stack we have a pushout

which implies that (7.4.4.1) is a pullback. Using again the fact that ( ) alg commutes with limits and invoking the induction hypothesis, we see that we are reduced to prove that the canonical map

is an equivalence. Let q : t n X ⇥ B(D n k (r)) ! t n X be the natural projection. Then

Analytic S 1 -algebras

Recall that we denote by S 1 the (derived) algebraic stack S 1 : (dA↵ afp k )op ! S obtained as sheafification of the constant presheaf associated to the space S 1 . We denote by (S 1 ) an the (derived) analytic stack

) an : dAfd k op ! S obtained as the analytification of S 1 . Equivalently, (S 1 ) an is the derived analytic stack obtained as sheafification of the constant presheaf associated to the space S 1 . Notice that (S 1

) an is in fact underived and that ⇡ 0 ((S 1 ) an ) ' Sp(k).

In particular, Theorem 7.4.2.5 implies that an k ((S 1 ) an ) is a local nonconnective analytic ring.

Notation 7.4.5.1. We denote the local k-analytic ring an k ((S 1 ) an ) simply by an k (S 1

). Let X be any 1-topos. Let p ⇤ : X ⌧ S : p 1 be the canonical geometric morphism. We set an X (S 1

) := p 1 an k (S 1 ) 2 Str loc T nc an (k) (X).

Nonconnective analytic square-zero extensions

We now turn our attention to the second main character in the HKR theorem: the analytic split square-zero extension.

Using ?? we obtain an equivalence Mod k ' Sp(AnRing nc k ). We simply denote the forgetful functor With respect to this monoidal structure, the functor ⌦ 1 is lax monoidal. Using Theorem 7.3.7.5, we obtain a canonical map

which in general is not an equivalence. On the other hand, using the formal nonconnective context, the generic fiber transformation and Theorem 7. We now provide an alternative construction for the category "-AnRing nc k .

Lemma 7.4.7.3. The canonical morphism

)) alg is an equivalence.

Proof. Need relative Van Est.

Analytic HKR

Prove S 1 -equivariant HKR (define mixed analytic rings as comodules over the analytic split square-zero extension).

Our goal is to prove the following theorem: Proof. The existence of L is a consequence of the adjoint functor theorem and the fact that the comonad k[⌘] an b ⌦ k acts on A alg by A 7 ! A A [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]. We now remark that for every f : A ! B in AnRing k , we have a fiber sequence

Similarly, we have a fiber sequence Finally, we observe that there is a canonical map

induced by A ! Sym an A ( L an A [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]). This map induces a morphism

The above fiber sequences imply that this morphism is an equivalence. Hence the functor A 7 ! Sym an A ( L an A [START_REF] António | Moduli of p-adic representations of a profinite group[END_REF]) is left adjoint to A 7 ! k[⌘] an b ⌦ k A. This completes the proof.