Michel Gros

Bernard Le Stum

Adolfo Quirós

Twisted differential operators of negative level and prismatic crystals

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Introduction

Recently Barghav Bhatt and Peter Scholze have introduced in [START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF] the beautiful notion of a prism. It is built on the concept of a thickening which goes back to Alexander Grothendieck in its modern form. A thickening of a ring B (always commutative) is simply another ring B together with a surjective map B Ñ B. Alternatively, we may consider the couple pB, Jq where J denotes the kernel of the surjection so that B " B{J. The idea behind any theory using this concept is to allow only some particular thickenings with some extra structure in order to get a better grasp on B, which is actually the main object of study even if it sometimes disappears from the picture. Grothendieck used infinitesimal thickenings (J nilpotent) in order to develop calculus over B and Berthelot extended the theory to positive characteristic p by using PD-thickenings (divided powers on J).

A prism is a particular kind of a thickening. For example, if we start from B " Z p rζs where ζ is a primitive pth root of unity, we may consider the thickening B " Z p rrq ´1ss obtained by sending q to ζ. The ideal J is then generated by the q-analog ppq q " 1 `q `¨¨¨`q p´1 of p. We will only consider here prisms over this base that we may call ppq q -prisms and we will stick to bounded prisms (a very light finiteness condition). More precisely, a prism (in this context) is a ppq q -torsion free pp, q´1q-adically complete Z p rrq´1ss-algebra B endowed with a δ-structure 1 satisfying δpqq " 0 (identifying q with its image in B) and such that B " B{ppq q B has bounded p 8 -torsion. An important example from p-adic Hodge theory is Fontaine's ring A inf :" W pO 5

Cp q with q " rζs (Teichmüller lifting of a sequence of pth roots of ζ) in which case A inf " O Cp . This applies more generally to perfectoid integral rings which then correspond to what are called perfect prisms.

Putting all infinitesimal thickenings together provides the infinitesimal site on which crystals are defined. Any crystal 2 (say on X smooth over S) gives rise to a module endowed with an integrable connection (on X{S) and this defines an equivalence of categories. Actually, in positive characteristic p, one uses PD-thickenings and obtain the crystalline site. A similar pattern occurs with prisms. In this article, we consider the following situation: we let R be a ppq q -prism and A a pp, q ´1q-adically complete R-algebra which admits a coordinate x. There exists a unique δ-structure on A such that δpxq " 0 and a unique endomorphism σ such that σpxq " qx. Then, one may consider the prismatic site of A{R and we will show in section 6 that a prismatic crystal gives rise to an Amodule endowed with what we call a twisted connection of level ´1. In general, a twisted connection of level ´m on an A-module M is an R-linear map satisfying the following Leibniz rule: @f P A, @s P M, ∇pf sq " pp m q q s b d q p m f `σp m pf q∇psq.

We will denote the corresponding category by MIC p´mq q pA{Rq. There exists also the notion of a q-PD-thickening giving rise to the q-crystalline site and we showed in [START_REF] Gros | Twisted differential operators and q-crystals[END_REF] that, if a is a q-PD-ideal in A, then a q-crystal on A{a gives rise to an A-module endowed with a twisted connection of level 0.

The main result of the present article is the following: we prove (theorem 4.8 and corollary 4.9) that, if A 1 denotes the frobenius pullback of A, then frobenius descent provides an equivalence of categories between A 1 -modules endowed with a twisted connection of level ´1 and A-modules endowed with a twisted connection of level 0 (when we focus on finitely presented topologically quasi-nilpotent objects). Moreover, as we expected in [START_REF] Gros | Sur une q-déformation locale de la théorie de Hodge nonabélienne en caractéristique positive[END_REF], section 6, this is related to prisms via a very general Cartier morphism C, whose definition is inspired by the proof of theorem 16.17 of [START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF], from the q-crystalline topos of a smooth formal scheme X to the prismatic topos of its frobenius pull back X 1 . When X " SpfpA{aq, in which case X 1 " SpfpA 1 {ppq q Aq, there exists a commutative diagram tppq q ´prismatic crystals on X 1 {Ru C ´1 / / tq´crystals on X {Ru MIC p´1q q pA 1 {Rq F ˚/ / MIC p0q q pA{Rq.

As explained above, the bottom map is an equivalence (on finitely presented topologically quasi-nilpotent objects) and, introducing flat topology considerations, we will prove in a forthcoming article that this is also the case for the vertical maps. Of course, this implies that the upper map is also an equivalence and we expect this to be true more generally for any smooth formal scheme X . This would be a twisted arithmetic version of the results obtained by Oyama in [START_REF] Oyama | PD Higgs crystals and Higgs cohomology in characteristics p[END_REF] and Xu in [START_REF] Xu | Lifting the Cartier transform of Ogus-Vologodsky modulo p n[END_REF].

Let us briefly describe the content of this article. In the first section, we introduce the ring of twisted divided polynomials of negative level and show in proposition 1.12 that it automatically inherits a δ-structure in the case of level ´1. In the second section, we explain the basics of twisted calculus on an adic ring and give a fundamental example. In section three, we develop twisted calculus of negative level. In section four, we introduce the level raising functor and show in theorem 4.8 that this is an equivalence when we move from level ´1 to level 0 (Frobenius descent). In section five, we recall some basic notions on prisms and show in corollary 5.6 that the ring of twisted divided polynomials of level minus one is the prismatic envelope of the polynomial ring for the symmetric δ-structure.

In the last section, we explain Cartier descent from prismatic crystals to q-PD-crystals and we show in proposition 6.9 that this is compatible with the raising level map of section four (which is an equivalence in this case).

Twisted divided powers of negative level

We recast here some results from [START_REF] Gros | Twisted divided powers and applications[END_REF] (see also section 2 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF]) and extend them to negative level.

We let R be any commutative ring. We fix some q P R and denote by pnq q P R the q-analog of an n P Z ě0 (see [START_REF] Le | On quantum integers and rationals[END_REF] for example for a presentation of the theory of q-analogs). We let A be any commutative R-algebra and fix some x P A. We also fix a prime p and some m P Z ě0 .

If y P A and n P Z ě0 , we will consider the twisted powers ξ pnqq,y :"

n´1 ź i"0
pξ `piq q yq P Arξs.

(1)

They form an alternative basis for the polynomial ring Arξs as a free A-module. Note however that multiplication of twisted powers is kind of tricky because ξ pn 1 qq,y ξ pn 2 qq,y " min pn 1 ,n 2 q ÿ i"0

p´1q i piq q !q ipi´1q 2 ˆn1 i ˙qˆn 2 i ˙qy i ξ pn 1 `n2 ´iqq,y ,
as shown in lemma 1.2 of [START_REF] Gros | Twisted divided powers and applications[END_REF]. We will need to understand how blowing-up and frobenius act on twisted powers and we can already notice the following:

Lemma 1.1. If y, z P A, then the blowing up Arξs Ñ Arωs, ξ Þ Ñ zω sends ξ pnqq,zy to z n ω pnqq,y .

Proof. We have ś n´1 i"0 pzω `piq q zyq " z n ś n´1 i"0 pω `piq q yq .

Later, we will assume that R is a δ-ring (see section 1 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF] for a short review) and in particular that R is endowed with a lifting of frobenius given by φpf q " f p `pδpf q. We will also assume that q has rank one which means that δpqq " 0 (so that φpqq " q p ). We will denote3 by

A 1 :" R φ Ô b R A
the frobenius pullback4 of A. It is easy to see that the semilinear morphism of rings

A Ñ A 1 , f Þ Ñ f 1 :" 1 φ Ô b f
has the following effect on twisted polynomials:

Arξs Ñ A 1 rξs, ξ pnqq,y Þ Ñ ξ pnq q p ,y 1
(with y P A and y 1 " 1 φ Ô b y).

Back to the general situation, the next result will follow from section 2 in [GLQ19]:

Proposition 1.2. Given y P A, there exists a unique natural multiplication on the free A-module Axξy q,y with basis tξ rnsq,y u nPZ ě0 turning the A-linear map Arξs Ñ Axξy q,y , ξ pnqq,y Þ Ñ pnq q !ξ rnsq,y into a morphism of rings.

Proof. Since this is the first time that we use the term "natural", we should make it precise (and leave it to the reader's imagination in the future): it means that if we are given a commutative diagram of commutative rings

q 1 _ R 1 / / A 1 y 1 _ q 2 R 2 / / A 2 y 2 , then the diagram ξ _ A 1 rξs / / A 1 xξy q 1 ,y 1 ξ rnsq 1 ,y 1 _ ξ A 2 rξs / / A 2 xξy q 2 ,y 2 ξ rnsq 2 ,y 2
is also commutative. Let us now prove our assertion. Existence follows from propositions 2.2 and 2.1 in [START_REF] Gros | Twisted divided powers and applications[END_REF]. For uniqueness, we may first replace A with some polynomial ring (in an infinite number of variables) over Zrqs, then replace Zrqs with its fraction field Qpqq and finally rely on proposition 2.1 of [START_REF] Gros | Twisted divided powers and applications[END_REF] again.

Definition 1.3. The ring Axξy q,y is the ring of twisted divided polynomials.

Note that, when pnq q ! is invertible in R, we have ξ rnsq,y " ξ pnqq,y pnq q ! so that ξ rnsq,y deserves the name of a twisted divided power.

We shall now show that blowing up and frobenius both extend to twisted divided powers.

Lemma 1.4. If y, z P A, then the blowing up ξ Þ Ñ zω extends naturally in a unique way to a morphism of A-algebras Axξy q,zy Ñ Axωy q,y , ξ rnsq,zy Þ Ñ z n ω rnsq,y .

Proof. First of all, the formula defines an A-linear map that extends the original blowing up. In order to show that it is a morphism of rings, we may actually assume that R " Qpqq in which case the question becomes trivial. The same argument can be used to show uniqueness.

On proves exactly in the same way the following:

Lemma 1.5. If R is a δ-ring with δpqq " 0 and A 1 denotes the frobenius pullback of A, then there exists a unique natural semilinear morphism of rings Axξy q,y Ñ A 1 xξy q p ,y 1 , ξ rnsq,y Þ Ñ ξ rns q p ,y 1 (with y P A and y 1 " 1 φ Ô b y).

Note that we actually obtain an isomorphism

R φ Ô b R Axξy q,y » A 1 xξy q p ,y 1 .
Recall that we fixed some parameter x P A (that we haven't used yet). We will then set y :" p1 ´qqx and simply write ξ pnqq for the twisted powers so that ξ pnqq :"

n´1 ź i"0 `ξ `p1 ´qi qx ˘P Arξs.
In this situation, we will also denote by Axξy q the ring of twisted divided polynomials and write ξ rnsq for the corresponding twisted divided powers.

We can replace q with q p m in order to obtain a new ring Axξy q p m but y needs then to be replaced by p1 ´qp m qx " pp m q q y in order to fit the pattern (for fixed x) so that Axξy q :" Axξy q,y and Axξy q p m :" Axξy q p m ,pp m qqy .

The new ingredient in our story is the ring obtained by changing q to q p m but keeping the same y (in which case we'd use ω rather than ξ as indeterminate):

Definition 1.6. Given m P Z ě0 , the ring of twisted divided polynomials of level ´m with respect to p is Axωy qp´mq :" Axωy q p m ,y with y :" p1 ´qqx.

We will then write ω tnuq :" ω rns q p m ,y and call them twisted divided powers of level ´m.

Remarks

1. The ring of twisted divided polynomials of level 0 is the same thing as the ring of twisted divided polynomials of definition 1.3 : Axωy qp0q " Axξy q (we will usually keep ξ as indeterminate in this situation).

2. If we write explicitly the multiplication on Axωy qp´mq , we get:

ω tn 1 uq ω tn 2 uq " ÿ 0ďiďmintn 1 ,n 2 u q p m ipi´1q 2 ˆn1 `n2 ´i n 1 ˙qp m ˆn1 i ˙qp m pq ´1q i x i ω tn 1 `n2 ´iuq .
We will now study blowing up and frobenius in this specific context.

Proposition 1.7. The blowing up ξ Þ Ñ pp m q q ω extends uniquely to a natural morphism of A-algebras Axξy q p m Ñ Axωy qp´mq , ξ rns q p m Þ Ñ pp m q n q ω tnuq .

Proof. Apply lemma 1.4 with q replaced by q p m and z replaced by pp m q q . Remarks 1. When pp m q q (resp. and pnq q p m !) is invertible in R, we may identify both rings in the proposition and we have ω tnuq " ξ rns q p m pp m q n q ˜resp. " ξ pnq q p m pp m q n q pnq q p m ! In particular, we will have ω " ξ{pp m q q and this is why it is important to use different letters.

2. More generally, there exists a natural morphism of A-algebras Axωy q p r p´m`rq Ñ Axωy qp´mq , ω tnu q p r Þ Ñ pp m´r q n q ω tnuq whenever r ě m.

3. As a consequence, if R is a δ-ring with δpqq " 0, then there exists a unique natural semilinear morphism of rings Axωy qp´m`1q Ñ A 1 xωy qp´mq , ω tnuq Þ Ñ ppq n q ω tnuq .

(using x 1 :" 1 φ Ô b x P A 1 :" R φ Ô b R A as fixed parameter on the right hand side).

Assume R is a δ-ring and A is a δ-R-algebra with both q and x of rank one.

Definition 1.8. The symmetric δ-structure on Arξs is defined by δpξq "

p´1 ÿ i"1 1 p ˆp i ˙xp´i ξ i .
It is easy to check that this is the unique δ-structure that extends the δ-structure of A and such that x `ξ also has rank one. We will also consider the corresponding relative frobenius

F : A 1 rξs " R φ Ô b R Arξs Ñ Arξs
which is then given by F pξq " px `ξq p ´xp .

We focus up to the end of the section on the case m " 1.

Theorem 1.9. Assume R is a δ-ring and A is a δ-R-algebra with both q and x of rank one. Then the relative frobenius F : A 1 rξs Ñ Arξs extends uniquely (through blowing up) to a natural morphism rF s : A 1 xωy qp´1q Ñ Axξy q .

Proof. We have to prove that there exists a unique natural morphism making the diagram

ξ _ A 1 rξs F / / Arξs ξ _ ppq q ω A 1 xωy qp´1q rF s / / Axξy q ξ.
commutative. This requires some work but this is done in section 7 of [START_REF] Gros | Twisted divided powers and applications[END_REF] where we showed that rF s is explicitly given by

ω tnu Þ Ñ pn ÿ i"n
b n,i x pn´i ξ ris with b n,i :" piq q ! pnq q p !ppq n q a n,i P R

(2) and a n,i :"

n ÿ j"0 p´1q n´j q ppn´jqpn´j´1q 2 ˆn j ˙qp ˆpj i ˙q. (3)
Definition 1.10. The morphism rF s is the divided frobenius.

Corollary 1.11. The symmetric δ-structure of Arξs extends uniquely in a natural way to Axξy q .

Proof. We define φ as the composition

ξ _ Axξy q φ ( ( ppq q ω A 1 xωy qp´1q rF s
/ / Axξy q and easily check that it solves the problem.

This last result also holds for Axωy qp´1q but we need to go back to our fancy formulas from [START_REF] Gros | Twisted divided powers and applications[END_REF] in order to prove it:

Proposition 1.12. Assume R is a δ-ring and A is a δ-R-algebra with both q and x of rank one. Then the symmetric δ-structure of Arξs extends uniquely (through blowing up) in a natural way to Axωy qp´1q .

Proof. It is actually sufficient to show that there exists a unique natural φ-structure on Axωy qp´1q : we may assume that R " Zrqs and A " Rrxs are p-torsion free and the notions of δ and φ-structures are then equivalent. In order to avoid confusions, let us denote by φ A : A Ñ A the frobenius of A. It follows from proposition 7.5 (and lemma 7.8 which shows that the sum actually starts at i " n) of [GLQ19] that the (absolute) frobenius on Arξs is the φ A -linear morphism given by

φ : Arξs Ñ Arξs, ξ pnq q p Þ Ñ pn ÿ i"n φpa n,i qx pn´i ξ piq q p
where the a n,i are defined in (3). We consider now the natural φ A -linear morphism of A-modules

φ : Axωy qp´1q Ñ Axωy qp´1q , ω tnuq Þ Ñ pn ÿ i"n ppq i q φpb n,i qx pn´i ω tiuq
where the b n,i are defined in (2). We want to show that this is a morphism of rings that reduces to the frobenius modulo p. Actually, since the blowing up map Arξs Ñ Axωy qp´1q , ξ pnq q p Þ Ñ pnq q p !ppq n q ω tnuq becomes an isomorphism when all q-analogs are invertible, it is sufficient to prove that the diagram

ξ pnq q p _ Arξs φ / /
Arξs ξ pnq q p _ pnq q p !ppq n q ω tnuq Axωy qp´1q φ / / Axωy qp´1q pnq q p !ppq n q ω tnuq is commutative. We compute the image of ξ pnq q p along both paths. On one hand, we find

pn ÿ i"n piq q p !ppq i q φpa n,i qx pn´i ω tiuq
and on the other hand

φ `pnq q p !ppq n q ˘pn ÿ i"n ppq i q φpb n,i qx pn´i ω tiuq .
In order to conclude, it is therefore sufficient to recall that φppiq q !q " piq q p ! and formula (2) then provides an equality piq q p !φpa n,i q " φ `pnq q p !ppq n q ˘φpb n,i q.

Example Assume p " 2. Then, we have φpξq " p1 `qqξ r2s `p1 `qqxξ and φpωq " p1 `qq 2 ω t2u `p1 `qqxω.

In particular, we see that there is no natural δ-structure (i.e. compatible with blowing up) on the polynomial ring Arωs in general: we do have φpξq P Arξs but φpωq R Arωs.

Remark The diagram

A 1 xωy qp´1q rF s / / φ ) ) Axξy q ξ _ A 1 xωy qp´1q ppq q ω
is commutative and may be used to define the δ-structure of Axωy qp´1q when A itself is a frobenius pullback (which will be the case in practice).

Twisted coordinate

This section is completely independent of the previous one. We recall some notions introduced in [LQ18] and give an important example of a situation where these notions apply. However, we work on adic rings as in [START_REF] Gros | Twisted differential operators and q-crystals[END_REF], and not merely on usual rings.

We assume that R is an adic ring (not necessarily complete) and that A is a complete adic R-algebra (not necessarily R-adic at this point). Adic rings are always assumed to admit a finitely generated ideal of definition. Unless otherwise specified, completion is always meant relative to the adic topology. We will not need it here but we could even assume that R and A are Huber rings as long as P A{R :" A b R A is also a Huber ring (multiplication on P A{R need not be continuous in general).

We denote by

p 1 : A Ñ P A{R , f Þ Ñ f b 1 and p 2 : A Ñ P A{R , f Þ Ñ 1 b f
the "projections" and by ∆ :

P A{R Ñ A, f b g Þ Ñ f g
the "diagonal map" (we use the -contravariant -geometric vocabulary and notations).

Unless otherwise specified, we will always consider P A{R as an A-module through p 1 and call p 2 the "Taylor map".

Assume now that A is a twisted R-algebra, which simply means that the R-algebra A is endowed with a continuous endomorphism σ. We extend σ to P A{R in an asymmetric way by the formula σpf b gq " σpf q b g. We let I A{R be the kernel of ∆ and define the twisted powers and twisted principal parts of order n:

I pn`1qσ A{R
:" I A{R σpI A{R q ¨¨¨σ n pI A{R q and P A{R,pnqσ :" p P A{R {I pn`1qσ A{R (where the hat (resp. the bar) indicates the completion (resp. the closure)).

Definition 2.1. Assume A is a twisted R-algebra and let x P A. Then x is a σ-coordinate if the canonical map

Arξs ďn Ñ P A{R,pnqσ , ξ Þ Ñ 1 b x ´x b 1 is bijective 5 for all n P Z ě0 .
It is actually convenient to extend σ to the polynomial ring Arξs by requiring that σpx `ξq " x `ξ and to introduce the twisted powers ξ pn`1qσ :" ξσpξq . . . σ n pξq P Arξs.

One easily sees that x is a σ-coordinate if and only if

Arξs{pξ pn`1qσ q » P A{R,pnqσ (isomorphism of A-algebras). In general, we also set

Ω A{R,σ :" I A{R {I p2qσ A{R
and we denote by d σ : A Ñ Ω A{R,σ the map induced by p 2 ´p1 . This A-module represents the σ-derivations D : A Ñ M , i.e. the R-linear maps satisfying the twisted Leibniz rule @f, g P A, Dpf gq " Dpf qg `σpf qDpgq.

When x is a σ-coordinate, Ω A{R,σ is a free module on the generator d σ x and we denote by B σ the corresponding derivation of A (so that d σ f " B σ pf qd σ x).

In order to make clear the relation with the notions introduced in section 1, note that if x is a σ-coordinate such that σpxq " qx with q P R, then ξ pn`1qq " ξ pn`1qσ . In this situation, we will use q rather than σ as index or prefix in all our notations so that we will write I pn`1qq A{R , P A{R,pnqq , Ω A{R,q , d q or B q , for example, and say q-coordinate or q-derivation (even if everything actually depends on σ).

Let us give an important example where there exists a q-coordinate (we already used these results without proofs in [START_REF] Gros | Twisted differential operators and q-crystals[END_REF]). Definition 2.2. A topologically étale coordinate x on A with respect to R is the image of an indeterminate X under some topologically étale (that is, formally étale and topologically finitely presented) map RrXs Ñ A.

Proposition 2.3. If q ´1 is topologically nilpotent and x is a topologically étale coordinate on A, then there exists a unique continuous endomorphism σ of A such that σpxq " qx. Moreover, x is a q-coordinate on A.

Proof. Since A is complete adic and q ´1 is topologically nilpotent, A is complete for the pq ´1q-adic topology. We may therefore assume that q ´1 is nilpotent. The existence and uniqueness of σ are trivially true when A " RrXs and x " X. In general, they follow from the commutativity of the diagram

RrXs σ / / RrXs / / A. A / / 4 4

A{pq ´1q

since RrXs Ñ A is formally étale. Now, since x is a topologically étale coordinate on A over R, it is also a topologically étale coordinate on A{pq ´1q over R{pq ´1q. Thus, modulo q ´1, the canonical map

Arξs{ξ pn`1q Ñ P A{R,pnqq (4)
reduces to an isomorphism pA{pq ´1qqrξs{ξ n`1 » P A{pq´1q,n . section using the commutative diagram RrX, ξs{ξ pn`1q / /

Arξs{ξ pn`1q

P A,pnqq / / 3 3 P A{pq´1q,n pA{pq ´1qqrξs{ξ n`1 » o o
and obtain injectivity.

One may also produce a frobenius on A using the same methods:

Proposition 2.4. Assume p is a topologically nilpotent prime in A and R is a φ-ring. If x is a topologically étale coordinate on A, then there exists a unique structure of φ-R-algebra on A with x of rank one (i.e. φpxq " x p ). Moreover, the relative frobenius

F : A 1 :" R φ Ô b R A Ñ A is free of rank p.
Proof. We proceed exactly as above and it is sufficient to prove existence and uniqueness of the relative frobenius F . First, we reduce to the case where p is nilpotent in A. In the case A " RrXs and x " X, there is nothing to do. In general, existence and uniqueness follow from the commutativity of the diagram

RrXs F / / RrXs / / A A 1 / / / / 4 4 A 1 {p F / / A{p,
in which the last map is the relative frobenius of A{p. It only remains to verify that F : A 1 Ñ A is free of rank p: actually, there exists an isomorphism A 1 rT s{pT p ´1 b xq » A which is obtained from the analog map when A " RrXs after pulling back along RrXs Ñ A 1 .

We can improve a little bit on the previous result (see also [START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF], lemma 2.18):

Proposition 2.5. Assume p is a topologically nilpotent prime in A and R is a δ-ring. If x is a topologically étale coordinate on A, then there exists a unique structure of δ-R-algebra on A with x of rank one (i.e. δpxq " 0).

Proof. There exists a unique structure of δ-R-algebra on RrXs such that δpXq " 0. Now, by definition (see section 1 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF]), a δ-structure on A is a section of the projection W 1 pAq Ñ A (where W 1 denote the Witt vectors of length two). Note that the kernel V 1 of this projection satisfies V 2 1 Ă pV 1 . By functoriality, there exists a commutative diagram

W 1 pRrXsq / / RrXs x x W 1 pAq / / A.
where the backward upper-map is the above trivial δ-structure of RrXs. Since V 1 is nilpotent modulo p and p is topologically nilpotent, the composite map

RrXs Ñ W 1 pRtXuq Ñ W 1 pAq
extends uniquely to A and defines a δ-structure on A with δpxq " 0.

Twisted calculus of negative level

We keep the same hypotheses as before: R is an adic ring and A is a complete adic Ralgebra, but we also fix6 , as in section 1, a prime p and some m P Z ě0 . We assume that A is a twisted R-algebra and that x P A is a q p m -coordinate (we will denote by σ p m the corresponding endomorphism of A even if it needs not be a power).

Definition 3.1. A twisted connection of level ´m on an

A-module M is an R-linear map ∇ : M Ñ M b Ω A{R,q p m such that @f P A, @s P M, ∇pf sq " pp m q q s b d q p m f `σp m pf q∇psq.
Intuitively, this is a twisted connection with a vertical pole: a twisted connection on M r1{pp m q q s which is defined over A.

We will denote7 by MIC p´mq q pA{Rq the category of A-modules endowed with a twisted connection of level ´m. The twisted de Rham cohomology of level ´m of M is defined as H 0 dR,qp´mq pM q " ker ∇ and H 1 dR,qp´mq pM q " coker ∇ or in the derived category as RΓ dR,qp´mq pM q " rM ∇ Ñ M b Ω A{R,q p m s.

Examples

1. When m " 0, we fall back onto the notion of a twisted connection from our previous articles: MIC p0q q pA{Rq " MIC q pA{Rq.

2. When q " 1, we remove the word "twisted" and we obtain a p m -connection in the sense of Deligne or Simpson as in definition 1.1 of [START_REF] Shiho | Notes on generalizations of local Ogus-Vologodsky correspondence[END_REF].

3. When pp m q q " 0 and σ p m " Id (which is then automatic in practice), a twisted connection of level ´m is the same thing as a Higgs field and does not depend on q, p or m anymore. This condition obviously holds when p " 0 but this also happens for example in the important case when q is a p m th root of unity.

4. The notion of a twisted connection of negative level is also related to the notion of a differential operator of finite radius that was studied in [START_REF] Le | Twisted calculus on affinoid algebras[END_REF] as well as, in the untwisted case, to the notion of a congruence level introduced in [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF]. See also corollary 1.3.15 in [START_REF] Kisin | Crystalline representations and F -crystals[END_REF], where Mark Kisin associates a λ-connection to a weakly admissible module.

Definition 3.2.

A twisted derivation of level ´m on an A-module M is an R-linear map θ : M Ñ M such that @f P A, @s P M, θpf sq " pp m q q B q p m pf qs `σp m pf qθpsq.

Of course, twisted connections and twisted derivations of level ´m correspond bijectively via ∇psq " θpsqd q p m x, but the later notion is often more convenient for computations.

Definition 3.3. The ring of twisted differential operators of level ´m on A{R is the free A-module D p´mq A{R,q on the generators B xny q

for n P Z ě0 with the commutation rules @f P A, B x1y q ˝f " pp m q q B q p m pf q `σp m pf qB x1y q and @n 1 , n 2 P Z ě0 , B xn 1 y q ˝Bxn 2 y q " B xn 1 `n2 y q .

Remarks 1. If we assume that x is also a q-coordinate and if we denote by D A{R,q the ring of twisted differential operators that we already considered in our previous articles, then there exists a natural A-linear morphism of rings

D p´mq A{R,q Ñ D A{R,q , B xny q Þ Ñ pp m q n q B n q p m . ( 5
)
More generally, we have

D p´mq A{R,q Ñ D p´m`rq A{R,q p r , B xny q Þ Ñ pp r q n q B xny q p r
for any r ď m if x is also a q p m´r -coordinate.

2. Giving a twisted connection (or a twisted derivation) of level ´m on an A-module M is equivalent to giving the structure of a D p´mq A{R,q -module. For the trivial twisted connection of level ´m on A, we can check that B xny q pf q " pp m q n q B n q p m pf q.

3. It is not difficult to see that RΓ dR,qp´mq pM q " RHom D p´mq A{R,q pA, M q.

Examples 1. In the case m " 0, we fall back onto the ring D A{R,q already considered in our previous articles.

2. In the case q " 1, then D p´mq A{R,q is the same thing as the ring D p´mq A{R of differential operators of level ´m introduced in definition 2.1 of [START_REF] Shiho | Notes on generalizations of local Ogus-Vologodsky correspondence[END_REF] (see also [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF]).

3. When A is ppq q -torsion-free and x is also a q-coordinate, the map (5) is injective, and we could then as well define D p´mq A{R,q as the A-subalgebra of D A{R,q p m generated by B x1y q :" pp m q q B q p m .

Recall that we introduced in definition 1.6 the ring Axωy qp´mq of twisted divided polynomials of level ´m. We will denote by I tn`1uq A{R the free A-module generated by ω tn 1 uq for n 1 ą n. This defines an ideal filtration on Axωy qp´mq . Lemma 3.4.

1. The blowing up ξ Þ Ñ pp m q q ω induces a canonical morphism of Aalgebras Arξs{pξ pn`1q q p m q Ñ Axωy qp´mq {I tn`1uq A{R .

2. If pp m q n q pnq q p m ! Ñ 0 in A, then the blowing up provides a canonical morphism of A-algebras Arrξss q p m :" lim Ð Ý Arξs{pξ pn`1q q p m q Ñ z Axωy qp´mq , ξ pnq q p m Þ Ñ pp m q n q pnq q p m !ω tnuq (recall that completion is meant with respect to the adic topology of A).

Proof. The first assertion follows from the fact that pp m q n q pnq q p m !ω tnuq P I tnuq A{R and the second one is obtained by taking the limit on both sides. More precisely, our hypothesis implies the existence of an ordered function n Þ Ñ rpnq and a canonical morphism

Arξs{pξ

pn`1q q p m q Ñ Axωy qp´mq {a rpnq Axωy qp´mq if a is an ideal of definition in A. Axωy qp´mq ) with a second structure of A-module commonly called the "right" structure (the usual one being the "left" structure).

2. It will follow from proposition 3.7 below that the twisted Taylor map of level ´m is explicitly given by f Þ Ñ ÿ B xiy q pf qω tiuq . This actually provides an alternative definition. Definition 3.6. A twisted differential operator of level ´m and order at most n is an A-linear map u : Axωy qp´mq {I tn`1uq A{R b 1 A M Ñ N (in which the b 1 indicates that we are using the "right" structure given by the twisted Taylor map of level ´m on the left hand side).

One can compose twisted differential operators of level ´m of different orders in the usual way using the diagonal maps

Axωy qp´mq {I tn 1 `n2 `1uq A{R ∆n 1 ,n 2 / / Axωy qp´mq {I tn 1 `1uq A{R b 1 A Axωy qp´mq {I tn 2 `1uq A{R ω tiuq / / ř i 1 `i2 "i ω ti 1 uq b ω ti 2 uq . ( 6
)
More precisely, the composition of two differential operators u 1 and u 2 of order at most n 1 and n 2 respectively is the differential operator of order at most n 1 `n2 defined by

u 1 u 2 :" u 1 ˝pId b 1 u 2 q ˝p∆ n 1 ,n 2 b 1 Idq.
Proposition 3.7. The twisted differential operators of level ´m of all orders from A to itself form a ring which is isomorphic to D p´mq A{R,q . More precisely, the basis tω tnuq u nPZ ě0 of Axωy qp´mq is "topologically" dual to the basis tB

xny q u nPZ ě0 of D p´mq A{R,q .
Proof. The point is to check that the bilinear map

D p´mq A{R,q ˆAxωy qp´mq {I tn`1uq A{R Ñ A, pB xny q , ω tku q q Þ Ñ " 1 if n " k 0 otherwise provides an isomorphism of rings D p´mq A{R,q » lim Ý Ñ n Hom A ´Axωy qp´mq {I tn`1uq A{R , A ¯.
Details are left to the reader.

Definition 3.8. A twisted Taylor structure of level ´m on an A-module M is a compatible family of A-linear maps

θ n : M Ñ M b A Axωy qp´mq {I tn`1uq A{R such that @n 1 , n 2 P Z ě0 , pθ n 1 b Id Axωy qp´mq {I tn 2 `1uq q ˝θn 2 " pId M b ∆ n 1 ,n 2 q ˝θn 1 `n2 .
Remarks 1. A twisted Taylor structure of level ´m is equivalent to a D p´mq A{R,q -module structure via the formula

θ n psq " n ÿ i"0 B xiy q s b ω tiuq .
This is therefore also equivalent to a twisted connection (or derivation) of level ´m.

2. One may define the Čech-Alexander cohomology of an A-module with a twisted Taylor structure of level ´m and show that this is isomorphic to de Rham cohomology.

3. As in definition 3.9 below, when ppq n q pnq q p ! Ñ 0, there exists an "hyper" equivalent to the notion of a twisted Taylor structure of level ´m with Axωy qp´mq {I tn`1uq A{R replaced with of z Axωy qp´mq .

There remains one more notion to investigate (we denote by ∆ n : Axωy qp´mq {I tn`1uq Ñ A the canonical map sending ω to 0). Definition 3.9. 1. A twisted stratification of level ´m on an A-module M is a compatible family of Axωy qp´mq {I tn`1uq -linear isomorphisms

n : Axωy qp´mq {I tn`1uq b 1 A M » M b A Axωy qp´mq {I tn`1uq
satisfying the normalization and cocycle conditions ∆ np n q " Id M and ∆ n,n p n q " p 1 p n q ˝p2 p n q.

2. If ppq n q pnq q p ! Ñ 0, then a twisted hyper-stratification of level ´m on an A-module M is an z Axωy qp´mq -linear isomorphism

: z Axωy qp´mq b 1 A M » M b A z Axωy qp´mq
satisfying the normalization and cocycle conditions.

We will denote by Strat p´mq q pA{Rq (resp. z Strat p´mq q pA{Rq) the category of twisted stratified (resp. hyper-stratified) modules of level ´m.

Remarks

1. The following are equivalent:

• a twisted connection (or derivation) of level m,

• a structure of a D p´mq A{R,q -module, • a twisted Taylor structure of level m,

• a twisted stratification of level m.

In particular, there exists an isomorphism of categories MIC p´mq q pA{Rq » Strat p´mq q pA{Rq.

Any twisted hyper-stratification of level ´m induces a twisted stratification of level

´m and we get a fully faithful functor

z Strat p´mq q pA{Rq Ñ Strat p´mq q pA{Rq » MIC p´mq q pA{Rq.
Proposition 3.10. When M is finitely presented, a hyper-stratification of level ´m is equivalent to a topologically quasi-nilpotent twisted connection of level ´m, meaning that @s P M, B xky psq Ñ 0.

Proof. Standard (see the proof of proposition 6.3 in [GSQ20], for example).

Level raising and Frobenius descent

We fix a prime p and a non-negative integer m. All commutative rings are assumed to be Zrqs pp,q´1q -algebras and are endowed with their pp, q ´1q-adic topology. Let R be a ppq q -torsion free δ-ring such that q :" q1 R has rank one and A a complete R-algebra with fixed topologically étale coordinate x (see definition 2.2).

We saw in propositions 2.3 and 2.5 that, in this situation, there exists on A a unique structure of twisted R-algebra such that x is a q-coordinate and a unique δ-structure such that x has rank one. Note that x is also a q p m -coordinate, i.e. a coordinate with respect to σ p m where σ denotes the endomorphism of A. Also, the construction applies as well to

A 1 :" R φ Ô p b R A with x replaced by x 1 :" 1 φ Ô p bx.
Finally, recall from proposition 2.4 that the relative frobenius F : A 1 Ñ A is free on the generators 1, x, . . . , x p´1 .

We will need below the following commutation rules: Lemma 4.1.

1. F ˝σp m " σ p m´1 ˝F 2. ppq q p m´1 x p´1 F ˝Bq p m " B q p m´1 ˝F .

Proof.

1. Since both p and q ´1 are topologically nilpotent and x 1 is a topologically étale coordinate on A 1 , there exists a unique morphism A 1 Ñ A sending x 1 to q p m x p . Now, we compute pF ˝σp m qpx 1 q " F pq p m x 1 q " q p m x p and pσ p m´1 ˝F qpx 1 q " σ p m´1 px p q " pσ p m´1 pxqq p " pq p m´1 xq p " q p m x p .

2. We first show that both maps are actually q p m -derivations from A 1 to A (seen as an A 1 -module via F ). In other words, they must both satisfy @f, g P A 1 , Dpf gq " F pf qDpgq `F pσ p m pgqqDpf q.

For the left hand side, this is automatic because we are composing a q p m -derivation of A 1 with an A 1 -linear map. For the right hand side, this follows from the first part of the lemma because pB q p m´1 ˝F qpf gq " B q p m´1 pF pf qF pgqq " F pf qB q p m´1 pF pgqq `σp m´1 pF pgqqB q p m´1 pF pf qq " F pf qpB q p m´1 ˝F qpgq `F pσ p m pgqqpB q p m´1 ˝F qpf q.

Since Ω A 1 {R,q p m is free on d q p m x 1 , a q p m -derivation of A 1 is determined by its value on x 1 . Now, we compute ppq q p m´1 x p´1 pF ˝Bq p m qpx 1 q " ppq q p m´1 x p´1 and pB q p m´1 ˝F qpx 1 q " B q p m´1 px p q " ppq q p m´1 x p´1 .

We can raise the level of a twisted derivation as follows:

Proposition 4.2. If θ 1 is a twisted derivation of level ´m on an A 1 -module M 1 , then there exists a unique derivation θ of level ´m `1 on M :"

A F Ô b A 1 M 1 such that @s P M 1 , θp1 b sq " x p´1 b θ 1 psq.
Proof. Necessarily, we will have for f P A and s P M 1 , θpf b sq " pp m´1 q q B q p m´1 pf q b s `xp´1 σ p m´1 pf q b θ 1 psq and we first need to show that this is well defined. On the one hand, we have for

f P A 1 , θp1 b f sq " x p´1 b θ 1 pf sq " x p´1 b pp m q q B q p m pf qs `xp´1 b σ p m pf qθ 1 psq " pp m q q x p´1 F pB q p m pf qq b s `xp´1 F pσ p m pf qq b θ 1 psq
and on the other hand, we have θpF pf q b sq " pp m´1 q q B q p m´1 pF pf qq b s `xp´1 σ p m´1 pF pf qq b θ 1 psq.

We may therefore rely on lemma 4.1 because pp m´1 q q ppq q p m´1 " pp m q q . It only remains to show that θ is a twisted derivation:

θpf g b sq " pp m´1 q q B q p m´1 pf gq b s `xp´1 σ p m´1 pf gq b θ 1 psq " pp m´1 q q B q p m´1 pf qg b s `pp m´1 q q σ p m´1 pf qB q p m´1 pgq b s `xp´1 σ p m´1 pf qσ p m´1 pgq b θ 1 psq " pp m´1 q q B q p m´1 pf qg b s `σp m´1 pf q ´pp m´1 q q B q p m´1 pgq b s `xp´1 σ p m´1 pgq b θ 1 psq " pp m´1 q q B q p m´1 pf qg b s `σp m´1 pf qθpg b sq.

Remarks

1. Alternatively, a twisted connection ∇ 1 of level ´m on M 1 gives rise to a twisted connection of level ´m `1 on M :"

A F Ô b A 1 M 1 .
2. This may also be interpreted in terms of D-modules but the functor does not come from a morphism between rings of differential operators. It will be necessary to consider hyper-stratifications to go further.

Definition 4.3. The functor

F ˚: MIC p´mq q pA 1 {Rq Ñ MIC p´m`1q q pA{Rq, pM 1 , θ 1 q Þ Ñ pM, θq (7)
is the level raising functor for twisted connections of negative level.

Remarks

1. We showed in corollary 8.9 of [START_REF] Gros | Twisted divided powers and applications[END_REF] that the level raising functor

F ˚: MIC p´1q q pA 1 {Rq Ñ MIC q pA{Rq, pM 1 , θ 1 q Þ Ñ pM, θq (8)
induces an equivalence on topologically quasi-nilpotent objects objects killed by ppq q . This does not extend to higher m.

2. Shiho proved in theorem 3.1 of [START_REF] Shiho | Notes on generalizations of local Ogus-Vologodsky correspondence[END_REF] that the level rising functor (8) is also an equivalence when q " 1 (the untwisted case) on topologically quasi-nilpotent objects. Again, this does not extend to higher m.

Up to the end of the section, we focus on the case m " 1.

The divided frobenius introduced in definition 1.10 is F -linear with respect to the "left" structure. One can show that it is also F -linear with respect to the "right" structure. More precisely, if we denote by rF s n : A 1 xωy qp´1q {I tn`1uq Ñ Axξy q {I rn`1s ´resp. y rF s : { A 1 xωy qp´1q Ñ z Axξy q the divided frobenius of order n (resp. the completed divided frobenius), then we have:

Lemma 4.4. The diagrams

A 1 θn / / F A 1 xωy qp´1q {I tn`1uq rF sn A θn / / Axξy q {I rn`1s
and

A 1 θ / / F { A 1 xωy qp´1q y rF s A θ / / z Axξy q are commutative.
Proof. Using definition 3.5, this follows from the commutativity of the various squares in the diagram (we only draw the completed case)

A 1 p 2 / / F P A 1 {R / / F lim Ð Ý p P A 1 {R { p I pn`1q q p A 1 {R » / / F A 1 rrξss q p / / F { A 1 xωy qp´1q y rF s A p 2 / / P A{R / / lim Ð Ý p P A{R { p I pn`1qq A{R » / / Arrξss q / / z Axξy q ,
the last one coming from theorem 1.9.

The following definitions therefore make sense: Definition 4.5.

1. The level raising functor

F ˚: Strat p´1q q pA 1 {Rq Ñ Strat p0q q pA{Rq
is the functor obtained by pulling back the stratification along the divided frobenius of finite orders.

The level raising functor

F ˚: z Strat p´1q q pA 1 {Rq Ñ z Strat p0q q pA{Rq
is the functor obtained by pulling back the hyper-stratification along the completed divided frobenius.

Remarks

1. Since rF spωq " x p´1 ξ mod ξ r2sq , one easily sees that the diagram Strat p´1q q pA 1 {Rq

F ˚/ / » Strat p0q q pA{Rq » MIC p´1q q pA 1 {Rq F ˚/ / MIC p0q q pA{Rq is commutative. 2. The diagram z Strat p´1q q pA 1 {Rq F ˚/ / z Strat p0q q pA{Rq Strat p´1q q pA 1 {Rq F ˚/ / Strat p0q q pA{Rq
is obviously also commutative.

Proposition 4.6. The divided frobenius rF s : A 1 xωy qp´1q Ñ Axξy q is free of degree p 2 .

Proof. Since F : A 1 Ñ A is free of degree p, it is sufficient to prove that the A-linearization

A F Ô b A 1 A 1 xωy qp´1q Ñ Axξy q
of rF s is free on 1, ξ, . . . , ξ p´1 . Thus, we have to show that trF spω tnuq qξ i u 0ďiăp,nPZ ě0 is a basis of Axξy q as an A-module. This follows from proposition 7.9 in [START_REF] Gros | Twisted divided powers and applications[END_REF] which gives the precise value of the leading coefficient

b n,pn " n ź k"1 p´1 ź i"1
pkp ´iq q P R în the explicit formula recalled in the proof of theorem 1.9.

Remarks

1. As a consequence, we see that the completed divided frobenius

y rF s : { A 1 xωy qp´1q Ñ z Axξy q
is also finite free and in particular faithfully flat. And the same is true for the morphism

y rF s p b y rF s : { A 1 xωy qp´1q p b A 1 { A 1 xωy qp´1q Ñ z Axξy q p b A z Axξy q .
2. The analogs at finite order are not true anymore.

Lemma 4.7. The composite map

Arξs Ñ A b R A A b A 1 A, ξ Þ Ñ 1 b x ´x b 1 (9)
extends uniquely to a morphism of δ-rings

u : Axξy q Ñ A b A 1 A (10)
and the diagram

A 1 xωy qp´1q rF s / / A 1 Axξy q u / / A b A 1 A (11)
is commutative (the upper map sends ω tnuq to 0).

Proof.

Since

φpξq P Arξs Þ Ñ 1 b x p ´xp b 1 P A b R A Þ Ñ 0 P A b A 1 A,
the first assertion follows from the universal property of Axξy q (theorem 3.6 in [START_REF] Gros | Twisted differential operators and q-crystals[END_REF]).

For the second assertion, we have to show that the map u sends rF spω tnuq q to 0 for n ą 0. By functoriality, we may clearly assume that R " Zrqs p,q´1 and A " Rrxs. It is then sufficient to show that pnq q p !ppq n q rF spω tnuq q is sent to 0 for n ą 0 but this is exactly φpξ pnq q which is a multiple of φpξq which itself, as we saw above, is sent to 0 in A b A 1 A.

Remark

The existence of the map u relies on the universal property of twisted divided powers and is not trivial at all: when p " 2, it sends ξ r2sq to x 2 p1 b 1q ´x b x and for odd p, it sends ξ rpsq to p1 ´qqp p´3 2 q q p pp ´1q q !

x p p1 b 1q `p´1 ÿ

i"1

p´1q i piq q !pp ´iq q ! q ipi´1q 2 x i b x p´i .
We can now prove Berthelot's frobenius descent (see theorem 2.3.6 in [START_REF] Berthelot | D-modules arithmétiques. II. Descente par Frobenius[END_REF]) in our situation (from level 0 to level ´1):

Theorem 4.8. Raising level provides an equivalence of categories

z Strat p´1q q pA 1 {Rq » z Strat p0q q pA{Rq.
Proof. Recall that a descent datum along F on an A-module M is an isomorphism

: A b A 1 M » M b A 1
A satisfying the cocycle and normalization conditions. If we denote by pA{A 1 q ´Mod the category of A-modules endowed with a descent data along F , then there exists an equivalence of categories

A 1 ´Mod » pA{A 1 q ´Mod, M 1 Þ Ñ pA b A 1 M 1 , canq,
where can denotes the canonical descent datum ([Sta19, Section 023F]). This happens because F is faithfully flat (descent is effective). On the other hand, pulling back an hyper-q-stratification along the map

u : Axξy q Ñ A b A 1 A,
provides a descent datum and we obtain a functor z Strat p0q q pA{Rq Ñ pA{A 1 q ´Mod.

Diagram (11) provides us with a commutative diagram of functors

z Strat p´1q q pA 1 {Rq / / A 1 ´Mod » z Strat p0q q pA{Rq / / pA{A 1 q ´Mod. Now, let M 1 , N 1 P z Strat p´1q q
pA 1 {Rq and

u : M :" A b A 1 M 1 Ñ N :" A b A 1 N 1
be a morphism in z Strat p0q q pA{Rq. The map u induces a morphism in pA{A 1 q ´Mod. Since descent along F is effective, it comes from a unique A 1 -linear map u 1 : M 1 Ñ N 1 . Moreover, since y rF s is faithful, the diagram

{ A 1 xωy qp´1q b 1 A 1 M 1 / / 1bu 1 M 1 b A 1 { A 1 xωy qp´1q u 1 b1 { A 1 xωy qp´1q b 1 A 1 N 1 / / N 1 b A 1 { A 1 xωy qp´1q is commutative. It follows that u 1 is actually a morphism in z Strat p´1q q
pA 1 {Rq and our functor is therefore fully faithful. Now, if M P z Strat p0q q pA{Rq, it has a descent datum along F and comes therefore from an A 1 -module M 1 . Since y rF s is fully faithful, the hyperq-stratification of M comes from a a unique isomorphism

{ A 1 xωy qp´1q b 1 A 1 M 1 » M 1 b A 1 { A 1 xωy qp´1q .
The cocycle and normalization conditions then follow from the faithfulness of y rF s p b y rF s.

We may now state the twisted Simpson correspondence in this setting (generalizing corollary 8.9 of [START_REF] Gros | Twisted divided powers and applications[END_REF]):

Corollary 4.9. Raising level induces an equivalence between locally quasi-nilpotent finitely presented A 1 -modules endowed with a twisted connection of level ´1 and locally quasinilpotent finitely presented A-modules endowed with a twisted connection of level 0.

Prisms and twisted divided powers of negative level

The content of this section is quite similar to sections 3, 4 and 5 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF], but the proofs are usually easier.

We let R be a δ-ring with fixed rank one element q and we assume that R is actually a Zrqs pp,q´1q -algebra. It does not really matter at this point, but we will also assume that R is ppq q -torsion free. All R-algebras (or modules) are implicitly endowed with their pp, q ´1q-adic topology and completion will always be meant with respect to this topology. Finally, we will use here our notations from [START_REF] Gros | Twisted differential operators and q-crystals[END_REF] and denote by A δ the δ-envelope of an R-algebra A and by I δ the δ-envelope of an ideal I (so that, for example, when A " Rrxs and I " pxq, we have A δ " Rrx 0 , x 1 , . . .s and pxq δ " px 0 , x 1 , . . .q).

We recall that, by definition, a δ-pair over R is a couple pB, Jq made of a δ-R-algebra B and an ideal J Ă B. With the obvious morphisms, δ-pairs form a category. When J " pdq is a principal ideal, we will simply write pB, dq. We introduce now the algebraic version of a prismatic envelope:

Definition 5.1. The ppq q -envelope of a δ-pair pB, Jq (if it exists) is a δ-pair which is universal for morphisms from pB, Jq to ppq q -torsion free δ-pairs of the form pB 1 , ppq q q. This means that there exists a ppq q -torsion free δ-ring that we will usually denote by BrJ{ppq q s δ and a morphism B Ñ BrJ{ppq q s δ of δ-rings such that JBrJ{ppq q s δ Ă ppq q BrJ{ppq q s δ satisfying the following universal property: any morphism of δ-rings B Ñ B 1 where B 1 is ppq q -torsion free and JB 1 Ă ppq q B 1 factors uniquely through BrJ{ppq q s δ .

Remarks

1. Let pB, Jq be a δ-pair and b Ă B a δ-ideal (meaning that it is stable under δ). Assume that pB, Jq has a ppq q -envelope BrJ{ppq q s δ . Then, if the quotient ring BrJ{ppq q s δ {bBrJ{ppq q s δ is ppq q -torsion free, this is the ppq q -envelope of pB{b, pJ `bq{bq.

2.

Let tpB e , J e qu ePE be a commutative diagram of δ-pairs all having a ppq q -envelope B e rJ e {ppq q s δ , B :" lim Ý Ñ B e and J " ř J e B. If lim Ý Ñ B e rJ e {ppq q s δ is ppq q -torsion free, then this is the pqq p -envelope of pB, Jq.

Examples

1. If B is ppq q -torsion free and J Ă ppq q B, then BrJ{ppq q s δ " B.

2. If J " B, then BrJ{ppq q s δ " B " t1{pnq q u nPZ ě0 ‰ .

3. Let B :" Rrxs δ and J :" pxq δ . Then the ppq q -envelope of pB, Jq is Rrws δ with respect to the blowing up Rrxs Ñ Rrws, x Þ Ñ ppq q w. One can derive many other examples from this one.

4. Assume B " Rrxs, δpxq " 0 and p " 2. Then, δ ˆx p2q q ˙" q p2q q 2 ˆx p2q q ˙2 R R " x p2q q  (the ppq q -envelope is not noetherian in general).

Theorem 5.2. Let A be p-torsion free δ-R-algebra with fixed rank one element x. If we endow Arξs with the symmetric δ-structure (with respect to x), then the ppq q -envelope of pArξs, ξq is Axωy qp´1q .

Proof. This is analog to theorem 3.6 of [GSQ20] and we will only give a sketch of the proof. We may assume if we wish that R " Zrqs pp,q´1q and A " Rrxs. The point is to show that, if we let v n " ś pδ r pωqq ar where n " ř a r p r denotes the p-adic expansion of a non-negative integer, then tv n u nPZ ě0 is an alternative basis for Axωy qp´1q . The universal property will then be automatic. It is clearly sufficient to prove that δ r pωq " c r ω tp r uq mod F p r ´1 with c r P R ˆ (12) where F n denotes the A-submodule generated by ω tkuq for k ď n.

Let us first show that δ ´ωtp r uq ¯" d r ω tp r`1 uq and φ ´ωtp r uq ¯" e r ω tp r`1 uq mod F p r`1 ´1 with d r P R ˆand e r R R ˆ. It is not difficult to see that there exists such congruences with d r , e r P R. Since R is a local ring with maximal ideal pp, q ´1q, it is then sufficient to prove the conditions d r P R ˆand e r R R ˆmodulo pp, q ´1q. Since pq ´1q is a δ-ideal, we may therefore assume that q " 1 (so that R " Z p ). We can do the following computations

φ ´ωtp r u ¯" p p r pp´1q p r`1 ! p r ! ω tp r`1 u , ´ωtp r u ¯p " p r`1 ! pp r !q p ω tp r`1 u mod F p r`1 ´1
and check that the valuations of the coefficients are p r`1 ą 1 (which shows that v p pe r q ą 0) and 1 respectively. Finally, since

δ ´ωtp r u ¯" φ `ωtp r u ˘´`ω tp r u ˘p p ,
we see that v p pd r q " 0.

We can now prove congruence (12) by induction as follows:

δ r`1 pωq " δ ´cr ω tp r uq ¯mod F p r`1 ´1

" c p r δ ´ωtp r uq ¯`δpc r qφ ´ωtp r uq ¯mod F p r`1 ´1

" pc p r d r `δpc r qe r qω tp r`1 uq mod F p r`1 ´1 " c r`1 ω tp r`1 uq mod F p r`1 ´1. Definition 5.3. An R-algebra B is said to be bounded (with respect to ppq q ) if B is ppq qtorsion free and B{ppq q B has bounded p 8 -torsion. We call a δ-pair pB, Jq bounded when B is bounded.

We assume from now on that R itself is bounded. We may then recall the following fundamental definition: Definition 5.4 (Bhatt-Scholze). A bounded prism over ´p R, ppq q ¯(we will say a ppq qprism over R) is a δ-pair over R of the form pB, ppq q q where B is complete and bounded.

Remarks

1. There exists a more general notion of prism or bounded prism that we will not consider here.

4. If B is a ppq q -prism, then the ideal ppq q B is closed in B, or equivalently B is p-adically complete (proposition 4.3 of [GSQ20]).

5. If B is a bounded δ-R-algebra, then ´p B, ppq q ¯is a ppq q -prism over R (proposition 4.3 of [GSQ20] again).

Definition 5.5. The (bounded) prismatic envelope of a δ-pair pB, Jq over R is a ppq qprism which is universal for morphisms from pB, Jq to ppq q -prisms over R.

Remarks

1. There exists a more general notion of prismatic envelope for non bounded prisms but it is then necessary to use derived completions.

2. If pB, Jq has a ppq q -envelope which is bounded, then the completion of this ppq qenvelope is the prismatic envelope of pB, Jq. Note hat B is automatically bounded when B is flat (over R).

3. Prismatic envelopes are usually quite hard to compute (but see corollary 5.6 just below).

Proposition 5.2 has the following consequence:

Corollary 5.6. Let A be bounded δ-R-algebra with fixed rank one element x. If we endow Arξs with the symmetric δ-structure (with respect to x), then the prismatic envelope of pArξs, pξqq is z Axωy qp´1q .

Finally, we also have the following:

Proposition 5.7. Let A be a complete bounded R-algebra with topologically étale coordinate x. Then z Axωy qp´1q is the prismatic envelope of pP A{R , I A{R q.

Proof. Using corollary 5.6, this is completely analog to the proof of theorem 5.2 in [START_REF] Gros | Twisted differential operators and q-crystals[END_REF].

Cartier descent and prismatic crystals

We keep the same notations as before: R is a δ-ring, q P R satisfies δpqq " 0 and we assume that R is a Zrqs pp,q´1q -algebra. We also assume now that R is bounded (definition 5.3) so that p p R, ppq q q is a (bounded) prism.

Definition 6.1. If X is a p-adic formal scheme8 over R{ppq q , then the ppq q -prismatic site of X {R is the category opposite9 to the category of all ppq q -prisms B over R endowed with a morphism SpfpB{ppq q q Ñ X over R{ppq q .

We will denote this category by ppq q ´PRISpX {Rq and endow it with the coarse topology. Also, when X " SpfpSq is affine, we will write S{R instead of X {R.

Remarks

1. Assume S is a complete p-adic R{ppq q -algebra. Then an object of ppq q ´PRISpS{Rq is essentially a commutative diagram

R / / B R{ppq q S / / B{ppq q
where B is a complete bounded δ-R-algebra. Since R is fixed, we can even simply use the one-line diagram S Ñ B{ppq q B.

2. One may define in general the prismatic site PRIS which is the category opposite to the category of all bounded prisms (that we did not define here).

3. There is no final object in PRIS but we may consider the slice category ppq q ´PRIS {R of all prisms over p p R, ppq q q. This category is a lot simpler and actually (anti) equivalent to the category of all complete bounded δ-R-algebras B.

4. We may fiber the category ppq q ´PRIS {R over the category of p-adic R{ppq q -formal schemes, and then the category ppq q ´PRISpX {Rq is exactly the fiber over X (but there is no final objet anymore). 5. Our category ppq q ´PRISpX {Rq is the same as (or more precisely the opposite to) the category pX {Rq ∆ in definition 4.1 of [START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF]. We choose to stick to the classical notations.

Proposition 6.2. Let A be a complete R-algebra with rank one topologically étale coordinate x and A :" A{ppq q . We denote by S :" Rrws δ and we endow

U :" S p b R A p» z Arws δ q
with the unique δ-structure such that δpxq " ppq q w. Then U is a covering of (the final object of the topos associated to) ppq q ´PRISpA{Rq.

Proof. This follows from various universal properties as in theorem 6.6 of [START_REF] Berthelot | Notes on crystalline cohomology[END_REF] for example (see also proposition 7.4 of [START_REF] Gros | Twisted divided powers and applications[END_REF]).

Remark We will show in a forthcoming article that A itself is a covering of the final object if we use the flat topology instead of the coarse topology.

A sheaf on ppq q ´PRISpX {Rq is given by a family of sets E B for each B P ppq q ´PRISpX {Rq and a compatible family of transition maps Note that, as we have already remarked, it is usually very hard to compute C. We will denote by z Strat p´1q q pB{Rq the category of B-modules endowed with a twisted hyperstratification of level ´1. Proposition 6.5. Let X be a formal R{ppq q -scheme, B P PRISpX {Rq and C the prismatic envelope of the diagonal in B b R B. If E is a ppq q -prismatic crystal on X {R, then there exists a unique twisted hyper-stratification of level ´1 on E B given by

: C b 1 B E B » E C » E B b B C
Proof. The cocycle and normalization conditions are automatic by functoriality.

This defines a functor tppq q ´prismatic crystals on X {Ru Ñ z Strat p´1q q pB{Rq.

As a consequence, we see that the notion of a prismatic crystal is closely related to the constructions made in section 3: Corollary 6.6. Let A be a complete R-algebra with rank one topologically étale coordinate x and A " A{ppq q . Then, there exists a functor tppq q ´prismatic crystals on A{Ru Ñ z Strat p´1q q pA{Rq Ñ MIC p´1q q pA{Rq sending E to E A .

Remark We will show in a forthcoming article that the first functor is an equivalence, and consequently, that the composite map is also an equivalence if we restrict to locally finitely presented crystals and finitely presented modules with a topologically quasi-nilpotent connection.

Corollary 6.7. Let A be a complete R-algebra with rank one topologically étale coordinate

x. We denote by S :" Rrws δ and we endow U :" S p b R A p» z Arws δ q with the unique δ-structure such that δpxq " ppq q w. Then there exists an equivalence of categories tppq q ´prismatic crystals on A{Ru » z Strat p´1q q pU {Rq Example Let pR, rq Ñ pA, aq be a morphism of q-PD-pairs with A complete and bounded and a closed in A. If X :" SpfpA{aq and we let A 1 :" R φ Ô p b R A, then X 1 " SpfpA 1 {ppq q q and we have

C X {R ¨R / / A A{a A{a ‹ ‹ ' " ¨R / / A
A 1 {ppq q F / / A{ppq q ‹ ‹ ' in which F denotes as usual the relative frobenius of A.

Proposition 6.9. Let A be a complete R-algebra with rank one topologically étale coordinate x and a a closed q-PD-ideal in A. If A :" A{a and A 1 :" A 1 {ppq q with Proof. There exists a (vertical) morphism of ppq q -prisms on A 1 {R:

A 1 :" R φ Ô p b R A,
A 1 {ppq q A 1 {ppq q F A 1 o o o o F A 1 {ppq q F / / A{ppq q A o oo o
It follows that if E is a ppq q -prismatic crystal on A 1 {R, then we have

pC ´1 A{R Eq A " E A » A F Ô b A 1 E A 1 .
In the same way, the divided frobenius rF s : Axωy qp´1q Ñ Axξy p induces a morphism of ppq q -prisms on A 1 {R. It follows that the stratification on E A 1 comes from the q-crystalline structure of C ´1 A{R E.

Remarks

1. We proved in theorem 4.8 that the level rising map F ˚in the bottom of diagram ( 13) is an equivalence.

2. We will show in a forthcoming article that both vertical maps are an equivalence.

As a consequence, we will obtain for free that the upper map is also an equivalence.

3. One may hope to prove more generally that the Cartier morphism of definition 6.8 is always an equivalence of topos much as in [START_REF] Oyama | PD Higgs crystals and Higgs cohomology in characteristics p[END_REF] or [START_REF] Xu | Lifting the Cartier transform of Ogus-Vologodsky modulo p n[END_REF].

  E B Ñ E B 1 for each morphism B Ñ B 1 . The structural sheaf O X {R corresponds to the case E B " B and an O X {R -module is given by a family of B-modules E B together with (linear) transition mapsC b B E B Ñ E C . Definition 6.3. A ppq q -prismatic crystal on X {R is a sheaf of O X {R -modules on ppq q ´PRISpX {Rq with bijective transition maps C b B E B » E C .Remark Alternatively, this means thatO X {R b ΓpX {R,O X {R q ΓpX {R, Eq » E.Definition 3.9 can be generalized (it follows from proposition 5.7 that this is indeed a generalization): Definition 6.4. Let B be a complete bounded δ-R-algebra and C the prismatic envelope of the diagonal in B b R B. A twisted hyper-stratification of level ´1 on a B-module M is a C-linear isomorphism : C b 1 B M » M b B C satisfying the usual normalization and cocycle conditions.

  Remarks1. The twisted Taylor map of level ´m is a morphism of R-algebras that endows Axωy qp´mq {I

	tn`1uq A{R	(resp. z

Definition 3.5.

1. The twisted Taylor map level ´m and order n of A{R is the composite

θ n : A p 2 / / P A{R / / p P A{R { p I pn`1q q p m A{R » Arξs{pξ pn`1q q p m q / / Axωy qp´mq {I tn`1uq A{R .

2. If pp m q n q pnq q p m ! Ñ 0 in A, then the twisted Taylor map of level ´m of A{R is the composite map θ : A p 2 / / P A{R / / lim Ð Ý p P A{R { p I pn`1q q p m A{R » / / Arrξss q p m / / z Axωy qp´mq .

  then the diagram tppq q ´prismatic crystals on A 1 {Ru

				C	´1 A{R / / tq´crystals on A{Ru	(13)
	z Strat	p´1q q	pA 1 {Rq	F ˚/ / z Strat p0q q pA{Rq
	is commutative.

Later we will write A 1 :" R φ Ô p b R A but the topology is discrete here.

We always use the geometric vocabulary and say "pullback" instead of "scalar extension".

This is the σ-analog of differential smoothness (of relative dimension one).

Actually, the whole story will only depend on the integer k :" p m but the terminology and the notations involve m.

Be careful that our notation is different from Shiho's[START_REF] Shiho | Notes on generalizations of local Ogus-Vologodsky correspondence[END_REF]. He use MIC pmq for connections of level ´m but we prefer to keep MIC pmq for connections of positive level.

In[START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF], it is only required that B is derived complete. Actually, when B is bounded, derived completeness is equivalent to completeness, and we always assume here that the prisms are bounded.

A ppq q -prism over R is completely determined by the δ-R-algebra B and we may simply say that the ring B is a ppq q -prism over R. However, it is fundamental to understand that B is actually a δ-thickening with respect to ppq q of B :" B{ppq q , which is somehow the important object.

We will only consider topologically smooth formal schemes -otherwise, the definition has to be modified.

We'd rather stay close to the geometric feeling.

* Supported by grant PGC2018-0953092-B-I00 (MCTU/AEI/FEDER, UE). frobenius lift.

Proof. Follows from proposition 6.2.

Remark Although this will follow from further investigation, we cannot prove directly that the functor z Strat p´1q q pU {Rq Ñ z Strat p´1q q pA{Rq induced by the morphism of δ-R-algebras π : U Ñ A, w Þ Ñ 0 is an equivalence. There exists an obvious section s of the morphism π but it is not a δ-morphism. Even more: if we denote by V :" { pU b R U qrωs δ {pppq q ω ´ξq δ the prismatic envelope of the diagonal in U b R U , then the section s does not extend naturally to a section of the morphism V Ñ Axωy qp´1q , w Þ Ñ 0.

Recall now that we may also consider the q-crystalline site q´CRIS as we did in section 7 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF] as well as its slice category q´CRIS {R when R is endowed with a q-PDideal r. More generally, when X is a formal R{r-scheme, there exists the q-crystalline site q´CRISpX {Rq. This is the category opposite to the category of complete bounded q-PDpairs (see definition 3.1 of [START_REF] Gros | Twisted differential operators and q-crystals[END_REF]) pB, Jq over R together with a morphism of formal R{r-schemes SpfpB{Jq Ñ X . There exists a corresponding notion of a q-crystal and, with A as in lemma 6.6 and a a closed q-PD-ideal containing rA, if E is a q-crystal on pA{aq{R, then we may endow E A with a twisted hyper-stratification of level 0.

Let us mention (as this will be needed in the next definition) that if pB, Jq is a q-PD-pair, then φpJq Ă ppq q B and the frobenius therefore induces a morphism φ : B{J Ñ B{ppq q . We may now recall from the proof of theorem 16.17 of [START_REF] Bhatt | Prisms and prismatic cohomology. prépublication[END_REF] the following construction (we use the notation r to denote the associated topos): Definition 6.8. Let pR, rq be a q-PD-pair, X a formal R{r-scheme and X 1 :" X p b R{r Õφ R{ppq q . Then the morphism of topos

induced by the functor that sends 1. a q-PD-pair pB, Jq to the prism pB, ppq q q, 2. a morphism SpfpB{Jq Ñ X to the linearization SpfpB{ppq q q Ñ X 1 of the composite map SpfpB{ppq q q Ñ SpfpB{Jq Ñ X , is the prismatic Cartier morphism of X {R.

When X " SpfpSq, we may also write C S{R for this morphism.

Remarks

1. If E is a ppq q -prismatic sheaf on X 1 {R and pB, Jq is a (complete bounded) q-PD-pair, then we simply have pC ´1

X {R Eq B " E B .

2. As a particular case, we see that C ´1 X {R O X 1 {R " O X {R so that C X {S is trivially a (flat) morphism of ringed topos. It is also clear that C ´1 X {R preserves crystals.