Yves Andre

Yves Andre Perfectoid

Y Andr

PERFECTOID SPACES AND THE HOMOLOGICAL CONJECTURES

Keywords: 1991 Mathematics Subject Classification. 13D22, 13H05, 14G20 homological conjectures, big Cohen-Macaulay algebra, perfectoid algebra, purity

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct summand conjecture are included. One then portrays the progresses made with these (and related) techniques on the so-called homological conjectures.

THE DIRECT SUMMAND CONJECTURE

1.1. Let R be a Noetherian (commutative) ring and S a finite ring extension, and let us consider the exact sequence of finitely generated R-modules

(1.1) 0 → R → S → S/R → 0.
When does this sequence split? Equivalently, when is R → S pure, i.e. remains injective after any base change? This holds for instance when R → S is flat, or when R is a normal Q-algebra, but not in general (the embedding of Q[x, y]/(xy) in its normal closure gives a counter-example, since it is no longer an embedding modulo x + y).

The direct summand conjecture, formulated by M. Hochster around 1969, claims that (1.1) splits whenever R is regular. Hochster proved it when R contains a field [START_REF] Hochster | Contracted ideals from integral extensions of regular rings[END_REF]. R. Heitmann proved it in dimension ≤ 3 [START_REF]The direct summand conjecture in dimension three[END_REF].

1.2. Recently, the author proved it in general [START_REF] André | La conjecture du facteur direct[END_REF]: This has many (non trivially) equivalent forms. One of them is that every ideal of a regular ring R is contracted from every finite (or integral) extension of R. Another (more indirect) equivalent form is the following statement, which settles a question raised by L. Gruson and M. Raynaud [28, 1.4.3] 1 :

1.2.
1.2.2. Theorem. Any integral extension of a Noetherian ring descends flatness of modules.

We will see one more equivalent form below in the framework of the so-called homological conjectures (4.2).

THE ROLE OF PERFECTOID SPACES

2.1. Some heuristics. After Hochster's work [START_REF]Canonical elements in local cohomology modules and the direct summand conjecture[END_REF], it is enough to prove the direct summand conjecture in the case when R is a complete unramified local regular ring of mixed characteristic (0, p) and perfect residue field k. By Cohen's structure theorem, one may thus assume R = W (k)[[x 1 , . . . , x n ]].

In characteristic p, all proofs of the direct summand conjecture use the Frobenius endomorphism F in some way. In mixed characteristic, R = W (k)[[x 1 , . . . , x n ]] carries a Frobenius-like endomorphism (acting as the canonical automorphism of W (k) and sending x i to x p i ), which however does not extend to general finite extensions S of R. To remedy this, p-adic Hodge theory suggests to "ramify deeply", by adjoining iterated p th roots of p, x 1 , . . . , x n . Doing this, one leaves the familiar shore of Noetherian commutative algebra for perfectoid geometry, recently introduced by P. Scholze [START_REF] Scholze | Perfectoid spaces[END_REF].

To begin with, W (k) is replaced by the non-Noetherian complete valuation ring [START_REF] Gabber | Almost ring theory[END_REF], which gives precise meaning to "up to p 1 p ∞ -torsion"; for instance, p

K o := W (k)[p 1 p ∞ ].
1 p ∞ -almost etaleness means that p 1 p ∞ Ω L o /K o = 0.
Actually, [START_REF] Gabber | Almost ring theory[END_REF] is much more general: it deals with modules over a commutative ring up to k-torsion, for some idempotent ideal k. Going beyond the case of a valuation ideal k will be crucial: beside "p 1 p ∞ -almost" modules, we will have to consider "(pg) 1 p ∞ -almost" modules for some "geometric" discriminant g.

Perfectoid notions.

In perfectoid geometry, one works with certain Banach 2 Kalgebras A. One denotes by A o the K o -subalgebra of power-bounded elements. One says that A is uniform if A o is bounded, and that A is perfectoid if it is uniform and

F : A o /p x →x p → A o /p is surjective. An example which plays a crucial role in the sequel is ∪i W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ]
, a deeply ramified avatar of R. Morphisms of perfectoid algebras A → B are continuous algebra homomorphisms (one then says that B is a perfectoid A-algebra).

Perfectoid algebras enjoy three fundamental stability properties [START_REF] Scholze | Perfectoid spaces[END_REF]:

2.2.
f g } o is p 1 p ∞ -almost isomorphic to A o ( f g ) 1 p ∞
for some approximations f , g of f, g which admit iterated p th -roots in A.

Finite etale extension. Any finite etale extension

B of A is perfectoid, and B o is a p 1 p ∞ -almost finite etale extension of A o .
This generalization of 2.1 to perfectoid algebras is Faltings's "almost purity theorem" [START_REF] Faltings | Almost étale extensions[END_REF] as revisited by Scholze [START_REF] Scholze | Perfectoid spaces[END_REF] and Kedlaya-Liu [START_REF] Kedlaya | Relative p-adic Hodge theory, I: Foundations[END_REF].

Let us explain how the second assertion of 2.2.3 follows from the first following [1, 3.4.2]. The idea is to reduce to the case when B is Galois over A with Galois group G, i.e. B G = A and B ⊗A B ∼ → G B. This implies B oG = A o . On the other hand, since B is a finitely generated projective Amodule, B ⊗A B = B ⊗A B, and one deduces from 2.2.1 (assuming B perfectoid) that

B o ⊗A o B o → G B o is a p 1 p ∞ -almost isomorphism.
To get rid of the completion, one passes modulo p m : B o /p m 2 here and in the sequel, one can work with any perfectoid field K (of mixed char. (0, p)), -i.e. complete, non-discretely valued, and such that F is surjective on K o /p. An extensive dictionary between the langage of commutative algebra and the language of non-archimedean functional analysis is presented in [1, 2.3.1].

is almost Galois over A o /p m , hence almost finite etale, and a variant of Grothendieck's "équivalence remarquable" [12, 5.3.27]

Let us consider the perfectoid algebra

A = ∪i W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ] and notice that A o = ∪i W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]] is
1 p ∞ (e ⊗ 1) = 0 in Ext 1 R (S/R, R) ⊗ R A o ∼ = Ext 1 A o ((S ⊗ R A o )/A o , A o ).
One concludes that e = 0 by the following general elementary lemma (applied to M = Re and K = p

1 p ∞ A o ): 2.3.1. Lemma. [2, 1.1.2] Let R be a local Noetherian ring, M a finitely generated R- module, A a faithfully flat R-algebra. Let K be an idempotent ideal of A such that K.M A = 0 and R ∩ K = 0. Then M = 0.

The perfectoid Abhyankar lemma. In the general case, S

⊗ R A is no longer etale over A: one must take into account a discriminant g ∈ R of S[ 1 p ] over R[ 1 p ]
. This suggests to try to generalize 2.2.3 to ramified extensions of perfectoid algebras. It turns out that this is possible, provided one extracts suitable roots of g in the spirit of Abhyankar's lemma. This leads to replace everywhere "p

Theorem. [1]

Let A be a perfectoid K-algebra, which contains a compatible system of p-power roots g

1 p j of some non-zero-divisor g ∈ A o . Let B be a finite etale A[ 1 g ]- algebra. Let B be the integral closure of g -1 p ∞ A in B , so that B[ 1 g ] = B . Then B is almost perfectoid, and for any n, B o /p m is (pg) 1 p ∞ -almost finite etale (hence (pg) 1 p ∞ -almost flat) over A o /p m .
(If g = 1, one may use [12, 5.3.27] again to conclude that B o is itself almost finite etale over A o and recover 2.2.3).

The basic idea is to look at the pro-system of algebras of functions A j := A{ p j g } on complements of tubular neighborhoods of the hypersurface g = 0 in the perfectoid space Spa(A, A o ), resp. at the pro-system

B j := B ⊗ A[ 1 g ] A{ p j g }. Each A j is perfectoid (2.2.
2), and each B j is finite etale over A j , hence perfectoid; moreover B jo is p 1 p ∞ -almost finite etale over A jo by almost purity (2.2.3). One can show that B o is isomorphic to lim B jo , and that the latter has the asserted properties, However, in the sequel, the identification of (lim B jo )[

: if R denotes A o /p r for short, A jo /p r is p 1 p ∞ -almost isomorphic to R j := R[( p j g ) 1 p ∞ ]; the key idea is that for j ≥ j + rp k , R j → R j factors through R jk := s≤ 1 p k
R( p j g ) s , so that lim R j ∼ = lim R jk ; on the other hand, the kernel and cokernel of R → R jk are killed by g raised to a power which tends to 0 when j, k → ∞). Passing to the limit r → ∞, it follows that lim A jo = g -1 p ∞ A, and this also holds under the weaker assumption that

A is almost perfectoid, cf. [1, 4.2.2]. 2) lim 1 (B jo /p r ) is (pg) 1 p ∞ -almost zero. The technique is similar to the one in 1), cf. [1, 4.4.1]. 3) lim(B jo /p) F → lim(B jo /p) is (pg) 1 p ∞ -almost surjective. Indeed, taking the limit of the exact sequence 0 → B jo /p p-1 p → B jo /p → B jo /p 1 p → 0, one deduces from 2) (for r = p-1 p ), that lim(B jo /p) → lim(B jo /p 1 p ) is (pg) 1 p ∞ -almost surjective; on the other hand, B jo /p 1 p → B jo /p is an isomorphism because B j is perfectoid.
4) B is almost perfectoid. From 3), it suffices to show that the natural map 2.5. Infinite Kummer extensions. In order to extend the strategy of 2.3 to the general case by means of the perfectoid Abhyankar lemma, one has first to adjoin to the perfectoid algebra ∪i W (k)[p

B o /p = (lim B jo )/p → lim(B jo /p) is a (pg) 1 p ∞ -almost isomorphism. It is easy to see that it is injec- tive [1, 2.
1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ] the iterated p th - roots of a discriminant g.
At finite level i, it seems very difficult to control the extension (W (k)[p

1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][g 1 p i , 1 p ]) o , which is bigger than W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][g 1 p i
], but things turn easier at the infinite level, thanks to the perfectoid theory.

2.5.1. Theorem. [2, 2.5.2] Let A be a perfectoid K-algebra, and let g ∈ A o be a non-zero divisor. Then for any n, A g

1 p ∞ o /p m is p 1 p ∞ -almost faithfully flat over A o /p m .
The basic idea is to add one variable T , consider the perfectoid algebra C := A T 1 p ∞ and look at the ind-system of algebras of functions C i := C{ T -g p i } on tubular neighborhoods of the hypersurface T = g in the perfectoid space Spa(C, C o ).

The proof involves three steps.

7) A g 1 p ∞ o is p 1 p ∞ -almost isomorphic to colimi C o i .
This is an easy consequence of the general fact that for any uniform Banach algebra B and f

∈ B o , colimiB{ f p i } o is p 1 p ∞ -almost isomorphic to (B/f B) o , cf. [1, 2.9.3].
8) C o contains a compatible system of p-power roots of some non-zero-divisor fi such that

C o i is p 1 p ∞ -almost isomorphic to colimj C U o /(p 1 p j U -f 1 p j i
). This is one instance of Scholze's approximation lemma [30, 6.7]; one may assume fi ≡ T -g mod p

1 p . 9) C U o /(p 1 p j U -f 1 p j i , p m
) is faithfully flat over A o /p m . One may replace p m by any positive power of p, e.g. p

1 p j+1 . Since C is perfectoid, there is gij ∈ C 1 p ∞ o such that g p j ij ≡ g mod p. Then f 1 p j i ≡ T 1 p j -gij mod p 1 p j+1 , and C U o /(p 1 p j U -f 1 p j i , p 1 p j+1 ) ∼ = (A o /p 1 p j+1 )[T 1 p ∞ , U ]/(T 1 p j -gij), a free A o /p 1 p j+1 -module.

Conclusion of the proof of the

direct summand conjecture. chooses g ∈ R such that S[ 1 pg ] is etale over R[ 1 pg ]. One then follows the argument of 2.3, replac- ing A = ∪i W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ] by the "infinite Kummer extension" A := ( ∪i W (k)[p 1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ]) g 1 p ∞ . One introduces the finite etale extension B := S ⊗ R A[ 1 g ] of A[ 1 g ], and one replaces B = S ⊗ R A in 2.3 by B := (lim B jo )[ 1 p ],
noting that B o is the S-algebra lim B jo . Translating Th. 2.4.1 and Th. 2.5.1 in terms of almost purity modulo p m , and reasoning as in 2.3 (using the same lemma), one obtains that (1.1) splits modulo p m for any n [2, §3]. A Mittag-Leffler argument (retractions of R/p m → S/p m form a torsor under an artinian R/p m -module [17, p. 30]) shows that (1.1) itself splits.

Derived version.

In [START_REF]On the direct summand conjecture and its derived variant[END_REF], B. Bhatt revisits this proof and proposes a variant, which differs in the analysis of the pro-system A jo /p r occurring in the proof of Th. 2.4.1: he strengthens step 1) by showing that the pro-system of kernels and cokernels of (A/p r ) j → (A jo /p r ) j is pro-isomorphic to a pro-system of (pg) 1 p ∞ -torsion modules; this allows to apply various functors before passing to the limit j → ∞, whence a gain in flexibility. More importantly, he obtains the following derived version of the direct summand conjecture, which had been conjectured by J. de Jong: 2.7.1. Theorem. [START_REF]On the direct summand conjecture and its derived variant[END_REF] Let R be a regular Noetherian ring, and f : X → Spec R be a proper surjective morphism. Then the map R → RΓ(X, O X ) splits in the derived category D(R).

EXISTENCE OF (BIG) COHEN-MACAULAY ALGEBRAS

3.1. Cohen-Macaulay rings, and Cohen-Macaulay algebras for non-Cohen-Macaulay rings. Let S be a local Noetherian ring with maximal ideal m and residue field k. Recall that a sequence x = (x 1 , . . . , x n ) in m is secant 3 if dim S/(x) = dim S -n, and regular if for every i, multiplication by x i is injective in S/(x 1 , . . . , x i-1 )S. Any regular sequence 3 following Bourkaki's terminology (for instance); it is also often called "part of a system of parameters", although gr x S may not be a polynomial ring in the "parameters" x i (it is if x is regular). is secant, and if the converse holds, S is said to be Cohen-Macaulay. Regular local rings have a secant sequence generating m, and are Cohen-Macaulay.

Cohen-Macaulay rings form the right setting for Serre duality and the use of local homological methods in algebraic geometry, and have many applications to algebraic combinatorics [START_REF] Bruns | Cohen-Macaulay rings[END_REF]. When confronted with a non-Cohen-Macaulay ring S, one may try two expedients:

1) Macaulayfication: construct a proper birational morphism X → Spec S such that all local rings of X are Cohen-Macaulay. This weak resolution of singularities, introduced by Faltings, has been established in general by T. Kawasaki [START_REF] Kawasaki | On Macaulayfication of Noetherian Schemes[END_REF]. Hovewer, secant sequences in S may not remain secant (hence not become regular) in the local rings of X; this motivates the second approach:

2) Construction of a Cohen-Macaulay algebra4 : an S-algebra C such that any secant sequence of S becomes regular in C, and mC = C.

The existence of Cohen-Macaulay algebras implies the direct summand conjecture: indeed, if C is a Cohen-Macaulay algebra for a finite extension S of a regular local ring R, it is also a Cohen-Macaulay R-algebra; this implies that R → C is faithfully flat, hence pure, and so is R → S.

Constructions of Cohen-Macaulay algebras.

The existence of a (big) Cohen-Macaulay algebra was established by Hochster and C. Huneke under the assumption that S contains a field [START_REF]-Applications of the existence of big Cohen-Macaulay algebras[END_REF]. One may assume that S is a complete local domain. In char. p > 0, one may then take C to be the total integral closure of S (i.e. the integral closure of S in an algebraic closure of its field of fractions). This is no longer true in the case of equal char. 0, which can nevertheless be treated by reduction to char. p >> 0 using ultraproduct techniques.

The remaining case of mixed characteristic was settled in [START_REF] André | La conjecture du facteur direct[END_REF], using the same perfectoid methods, so that one has: p ∞ -almost isomorphic to a faithfully flat R algebra modulo any power of p. From there, one deduces that the sequence (p, x1, . . . , xn) is "(pg)

1 p i ][[x 1 p i 1 , . . . , x 1 p i n ]][ 1 p ]) g 1 p ∞ o and
1 p ∞ - almost regular" in B o .
To get rid of "almost", Lemma 2.3.1 is no longer sufficient: instead, one uses Hochster's technique of monoidal modifications [START_REF]Big Cohen-Macaulay algebras in dimension three via Heitmann's theorem[END_REF] [START_REF]-Applications of the existence of big Cohen-Macaulay algebras[END_REF]. After m-completion, one gets a S-algebra C in which (p, x1, . . . , xn), as well as any other secant sequence of S, becomes regular.

Subsequently, using the tilting equivalence between perfectoid algebras in char. 0 and in char.p and applying Hochster's modifications in char.p rather than in char.0, K. Shimomoto [START_REF]Integral perfectoid big Cohen-Macaulay algebras via André's theorem[END_REF] shows that in Th. 3.2.1, in mixed characteristic, C can be taken to be perfectoid 5 . In particular, if S is regular, it admits a perfectoid faithfully flat algebra (one may speculate about the converse6 ). If regularity is omitted, Spec(Q[x, y]/(xy)) and its normalization provide a counterexample.

HOMOLOGICAL CONJECTURES

4.1. Origins from intersection theory. Under the influence of M. Auslander, D. Buchsbaum and J.-P. Serre, commutative algebra has shifted in the late 50s from the study of ideals of commutative rings to the homological study of modules (cf. their characterization of regular local rings by the existence of finite free resolutions for any finitely generated module, resp. for the residue field).

Serre proved that for any three prime ideals p, q, r of a regular local ring R such that r is a minimal prime of p + q, ht r ≤ ht p + ht q [START_REF] Serre | Algèbre locale, multiplicités[END_REF]. The special case r = m can be amplified: for any ideals I, J of R such that

I + J is m-primary, dim R/I + dim R/J ≤ dim R.
This is no longer true if R is not regular, and attempts to understand the general situation led to the so-called homological conjectures cf. [7, ch. 9], [START_REF]Homological conjectures, old and new[END_REF] The first "intersection conjecture" was proposed by C. Peskine and L. Szpiro [START_REF] Peskine | Dimension projective finie et cohomologie locale[END_REF], proved by them when R contains a field by reduction to char. p and Frobenius techniques, and later proved in general by P. Roberts using K-theoretic methods [START_REF] Roberts | Le théorème d'intersection[END_REF]. It states that if M, N are finitely generated R-modules such that M ⊗ N has finite length, then dim N ≤ pd M . It implies that R is Cohen-Macaulay if and only if there is an R-module M of finite length and finite projective dimension (in the spirit of Serre's characterization of regular rings, which is the case M = k), resp. if there is an R-module M of finite injective dimension.

Indeed, one may take N = R and deduce that dim R ≤ pd M ; by the Auslander-Buchsbaum formula, pd M = depth R -depth M , so that the inequality depth R ≤ dim R is an equality. The second assertion follows from the fact that id M = depth R [START_REF] Bass | On the ubiquity of Gorenstein rings[END_REF].

The "new intersection conjecture", also proved by Peskine, Szpiro [START_REF] Peskine | Dimension projective finie et cohomologie locale[END_REF] and Roberts [START_REF] Roberts | Le théorème d'intersection[END_REF], states that for any non exact complex F • of free R-modules concentrated in degrees [0, s] with finite length homology, s > dim R.

The "improved new intersection conjecture" is a variant due to E. Evans and P. Griffith [START_REF] Evans | The syzygy problem[END_REF], in which the condition on F • is "slightly" relaxed: the H i>0 are of finite length and there exists a primitive cyclic submodule of H 0 of finite length. They proved it, assuming the existence of (big) Cohen-Macaulay algebras 7 , and showed that it implies their "syzygy conjecture". In spite of appearances, the passage from the new intersection conjecture to its "improved" variant is no small step 8 : in fact, according to Hochster [START_REF]Homological conjectures, old and new[END_REF] and S. Dutta [START_REF] Dutta | On the canonical element conjecture[END_REF], the latter is equivalent to the direct summand conjecture. On the other hand, in the wake of the new intersection conjecture (and motivated by the McKay correspondence in dimension 3 and the "fact" that threefold flops induce equivalences of derived categories), T. Bridgeland and S. Iyengar obtained a refinement of Serre's criterion for regular rings assuming the existence of Cohen-Macaulay algebras [6, 2.4].

By Th. 1.2.1 and Th. 3.2.1, the improved new intersection conjecture and the Bridgeland-Iyengar criterion thus hold inconditionally: 4.2.1. Theorem. Let R be a Noetherian local ring and F • be a complex of finitely generated free R-modules concentrated in degree [0, s], such that H >0 (F • ) has finite length.

(1) If H 0 (F • ) contains a cyclic R-submodule of finite length not contained in mH 0 (F • ), then s ≥ dim R. (2) If H 0 (F • ) has finite length and contains k as a direct summand, and s = dim R, then R is regular.

And so does the syzygy conjecture:

4.3.2.

In [START_REF] Ma | Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers[END_REF], Ma and K. Schwede define and study perfectoid multiplier/test ideals in mixed characteristic, and use them to bound symbolic powers of ideals in regular domains in terms of ordinary powers: 4.3.3. Theorem. [START_REF] Ma | Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers[END_REF] Let R be a regular excellent Noetherian domain and let I ⊂ R be a radical ideal such that each minimal prime of I has height ≤ h. Then for every n, I (hn) ⊂ I n .

Here I (hn) denotes the ideal of elements of R which vanish generically to order hn at I. When R contains a field, the result was proved in [START_REF] Ein | Uniform bounds and symbolic powers on smooth varieties[END_REF] and [START_REF] Huneke | Tight closure, invariant theory, and the Brianc ¸on-Skoda theorem[END_REF].

4.3.

3. An efficient and unified way of dealing with questions related to the homological conjectures in char. p is provided by "tight closure theory", which has some flavor of almost ring theory. Using Th. 2.4.1 and Th. 2.5.1 above, Heitmann and Ma give evidence that the "extended plus closure" introduced in [START_REF] Heitmann | Extensions of plus closure[END_REF] is a good analog of tight closure theory in mixed characteristic [START_REF]Extended plus closure in complete local rings[END_REF].

  1. Theorem. (1.1) splits if R is regular.

1 p 1 p 1 p

 111 ∞ -almost" by "(pg) ∞ -almost", thereby extending the basic setting of almost ring theory beyond the usual situation of a non-discrete valuation ring. This also leads to introduce the notion of almost perfectoid algebra, where F : A o /p x →x p → A o /p is only assumed to be (pg) ∞ -almost surjective[1, 3.5.4].

1 p 1 pG B jo is a p 1 p

 111 [START_REF] Faltings | Almost étale extensions[END_REF].1], and on the other hand, the composition limF,j(B jo /p) = B o → (lim B jo )/p → lim(B jo /p) is almost surjective by 3). 5) B → B j factors through a g1 p ∞ -almost isomorphism B{ p j g } a ∼ = B j .The factorization comes from the fact that B j ∼ = B j { p j g }, and one constructs an almost inverse to B{ p j g } a → B j as follows (cf.[1, 4.4.4]): the integral closure of A o in B maps to the integral closure A jo in B j , which is p ∞ -almost B jo (by almost purity). Passing to the limit and inverting pg, one gets a morphismB δ → B[ 1 g ],and the sought for inverse is induced by δ ⊗ 1 A j . 6) for every m, B o /p m is (pg) ∞ -almost finite etale over A o /p m . This is the decisive step: how to keep track of the almost finite etaleness of B jo over A jo at the limit? As in 2.2.3, the idea is to reduce to the case when B is Galois over A[1 g ] with Galois group G, i.e. (B )G = A[ 1 g ] and B ⊗ A[ 1 g ] B ∼ → G B .It follows that B j is G-Galois over A j , and (as in 2.2.3) that B jo ⊗A jo B jo → ∞ -almost isomorphism. Passing to the limit, (and taking into account the facts that, by 4) and 5), B ⊗A B is an almost perfectoid algebra and B ⊗A B{ p j g } ∼ = B j ⊗A j B j , so that one can apply 1)), one concludes that A o → B oG andB o ⊗A o B o → G B o are (pg)1 p ∞ -almost isomorphisms. To get rid of the completion, one passes modulo p m : B o /p m is almost Galois over A o /p m , hence (pg) 1 p ∞ -almost finite etale (in constrast to 2.2.3, one cannot conclude that B o is (pg) 1 p ∞ -almost finite etale over A o since p may not belong to (pg) 1 p ∞ ).

3. 2 . 1 .

 21 Theorem. Any local Noetherian ring S admits a (big) Cohen-Macaulay algebra C. In the case of a complete local domain S of char (0, p) and perfect residue field k (to which one reduces), one proceeds as follows. Cohen's theorem allows to present S as a finite extension of R = W [[x1, . . . , xn]]. One first considers the R-algebra A := ( ∪iW (k)[p

3. 3 .

 3 Finite and fpqc covers. Since Cohen-Macaulay algebras for regular local rings are faithfully flat, Th. 3.2.1 implies [2]: 3.3.1. Theorem. Any finite cover of a regular scheme is dominated by a faithfully flat quasi-compact cover.

  .

4. 2 .

 2 Intersection conjectures. Let (R, m) be local Noetherian ring.

7 8 K

 8 if R is Cohen-Macaulay, the Buchsbaum-Eisenbud criterion gives a condition for the exactness of F• in terms of codimension of Fitting ideals of syzygies. In general, the same condition guarantees that F• ⊗ C is exact for any Cohen-Macaulay R-algebra C[7, 9.1.8]. -theoretic techniques failed to make the leap.

4. 2 . 2 .

 22 Theorem. Let R be a Noetherian local ring and M a finitely generated R-module of finite projective dimension s. Then for i ∈ {1, . . . , s -1}, the i-th syzygy module of M has rank ≥ i.4.3.Further work around the homological conjectures using perfectoid spaces.4.3.1.In[START_REF] Ma | Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic[END_REF], Heitmann and L. Ma show that Cohen-Macaulay algebras can be constructed in a way compatible with quotients S → S/p by primes of height one9 . Using arguments similar to Bhatt's derived techniques, they deduce the vanishing conjecture for maps of Tor[START_REF]-Applications of the existence of big Cohen-Macaulay algebras[END_REF]: 4.3.1. Theorem.[START_REF] Ma | Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic[END_REF] Let R → S → T be morphisms such that the composed map is a local morphism of mixed characteristic regular local rings, and S is a finite torsionfree extension of R. Then for every R-module M and every i, the map Tor R i (M, S) → Tor R i (M, T ) vanishes.They obtain the following corollary, which generalizes results by Hochster and J. Roberts, J.-F. Boutot et al.:4.3.2. Corollary.[START_REF] Ma | Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic[END_REF] Let R → S be a pure, local morphism, with S regular. Then R is pseudo-rational, hence Cohen-Macaulay.

  The valuation ring L o of any finite extension L of the field K[ 1

p ] satisfies: (2.1) F : L o /p x →x p → L o /p is surjective, and L o is p 1 p ∞ -almost finite etale over K o , this being understood in the context of almost ring theory, introduced by G. Faltings and developped by O. Gabber and L. Ramero

  allows to conclude that B o it itself almost finite etale over A o .

	2.3. Direct summand conjecture: the case when S[ 1 p ] is etale over R[ 1 p ]. Let us go
	back to the direct summand conjecture for R = W (k)[[x 1 , . . . , x n ]]. The special case
	when S[ 1 p ] is etale over R[ 1 p ] was settled by B. Bhatt [4] and by K. Shimomoto [33], using
	2.2.3. Here is a slightly different account, suitable to the sequel.

  a faithfully flat extension of R. By assumption B := S ⊗ R A is finite etale over A, hence by 2.2.3, B o is p

	1 p ∞ -almost finite etale, hence
	almost pure (in the sense: almost universally injective) over A o . A fortiori, (1.1) almost
	splits after tensoring with A o . In other words, if e ∈ Ext 1 R (S/R, R) denotes the class
	corresponding to (1.1), then p

  1 p ] with the integral closure of g -1 p ∞ A in B plays no role; changing notation, we will set B := (lim B jo )[ 1 p ], which is a uniform Banach algebra, and sktech the proof that B is almost perfectoid and that for every m, B o /p m is (pg)1 p ∞ -almost finite etale over A o /p m . For any r ∈ N[ 1 p ], lim(A jo /p r ) is (pg) 1 p ∞ -almost isomorphic to A o /p r .This is essentially Scholze's perfectoid version of Riemann's extension theorem [31, II.3.1] (hint

	The proof involves six steps.
	1)

since it would be too restrictive to impose that C is Noetherian, one often speaks of "big" Cohen-Macaulay algebra.

actually, Shimomoto's result is slightly weaker, but can be enhanced.

this is settled in forthcoming work by Bhatt, Ma, Iyangar.

weak functoriality of Cohen-Macaulay algebras in general has been since announced by the author (arXiv:1801.10010).


Yves Andre

Yves Andr

SINGULARITIES AND PERFECTOID GEOMETRY

Keywords: 1991 Mathematics Subject Classification. 13D22, 13H05, 14G20 big Cohen-Macaulay algebra, perfectoid algebra, singularities. 1

These notes give a quick overview of recent work by the speaker and of some subsequent works introducing perfectoid geometry into homological commutative algebra and singularity theory. The emphasis is on big Cohen-Macaulay algebras and applications. The progresses take place primarily in mixed characteristic, but sometimes provide a bridge between characteristic p and characteristic 0.

SINGULARITIES: RESOLUTION VERSUS HOMOLOGICAL STUDY

1.1. Introduction. In his second letter to Leibniz (1677), Newton answered his correspondent's query about the foundation of his methods in this way: "the foundation is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux".

This anagram was decoded as: Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa 1 , which is usually grossly translated as:

It is useful to solve differential equations.

History has largely done justice to this watchword. But some three centuries later, another watchword emerged:

It is (also) useful not to solve differential equations... but to study their structure.

Proponents of this approach were Grothendieck who, reinterpreting the classical resolvent as descent datum, gave birth to the crystalline viewpoint on differential modules; and the Kyoto school (Sato, Kashiwara) who, treating solutions and cosolutions on equal footing, gave birth to the homological study of differential modules known as algebraic analysis.

This pattern "resolution versus homological study" also occurs in the algebraic theory of singularities.

Viewed as nuisances, one tries to get rid of them, which leads to the theory of resolution of singularities. Viewed as jewels of commutative algebra, gardening and classifying them leads to the homological study of singularities.

1.2. Cohen-Macaulay versus non Cohen-Macaulay singularities. This is a natural dichotomy: Cohen-Macaulay (CM) singularities form a rather broad class with lots of interesting examples from various origins, and they share many nice properties: they satisfy Serre duality, they allow concrete calculations of syzygies etc.

Let us remind their definition, and their homological characterization (reference: [START_REF] Bruns | Cohen-Macaulay rings[END_REF]).

We adopt the viewpoint and language of commutative algebra, where spaces are replaced by the commutative rings of their functions, locality being reflected by the uniqueness of the maximal ideal. Let S be a (Noetherian) local ring with maximal ideal m. A sequence x = (x 1 , . . . , x n ) of elements of m is secant if S/xS has dimension dim S -n.

Let M be an S-module.

A sequence x is M -regular if M/(x 1 , . . . , x i-1 )M •xi → M/(x 1 , . . . , x i-1 )M is injective (i = 1, . . . , n), and M = xM . M is a Cohen-Macaulay module if any secant sequence is M -regular.
S is a Cohen-Macaulay ring if any secant sequence is S-regular. This is in particular the case if S is regular, i.e. if m is generated by some secant sequence. Moreover, if S is regular, an S-module M is Cohen-Macaulay if and only if it is faithfully flat.

One has the following characterizations: [Auslander-Serre] S is regular ⇔ every finite S-module has a finite free resolution ⇔ S/m has a finite free resolution.

[Peskine-Szpiro-Roberts] S is Cohen-Macaulay ⇔ some nonzero S-module of finite length has a finite free resolution.

Big Cohen-Macaulay algebras. What to do in front of a non-Cohen-Macaulay ring S?

A first attitude (initiated by Faltings) is to try to get rid of the problem, finding a Cohen-Macaulay resolution. Such a weak resolution of singularities is now known to exist in great generality: 1.3.1. Theorem (Kawasaki [START_REF] Kawasaki | On Macaulayfication of Noetherian Schemes[END_REF], Cesnavicius [START_REF] Cesnavicius | Macaulayfication of Noetherian schemes[END_REF]). Let S be a quasi-excellent noetherian ring. There exists a projective morphism Y → X = Spec S with Y CM, which is an isomorphism over the CM locus of X.

A second attitude, prompted by Hochster, is to look for (big) CM algebras: namely, an S-algebra T (not necessarily of finite type) which is a CM S-module.

In the first approach (CM resolution), any secant sequence on Y (i.e. in the local rings of Y ) is regular, but a secant sequence on X needs not remain secant on Y .

In the second approach, any secant sequence on X becomes regular on Y = Spec T , but a secant sequence on Y needs not be regular.

Big CM-algebras turn out to be very important tools in the homological study of singularities (and surprisingly, even in the study of a CM singularity, see 2.2 and 3.2 below). More precisely: 1) for any complete local ring S, there is a CM S-algebra T , 2) for any chain of local homomorphisms S 1 → . . . → S n of complete local domains, there is a compatible chain T 1 → . . . → T n of CM algebras for S 1 , . . . S n respectively.

BIG COHEN-MACAULAY ALGEBRAS IN MIXED CHARACTERISTIC VIA PERFECTOID

It turns out that 1) is equivalent to the following more geometric statement: 1') For any regular ring R and any finite extension S, there is an S-algebra T which is faithfully flat over R.

Both 1) and 2) were conjectured by Hochster and proved by him in equal characteristic (partly with Huneke).

2.2. Three consequences of Theorem 2.1.1 in commutative algebra.

The direct summand conjecture [Hochster '69]: any finite extension S of a regular ring R splits (as R-module).

Another direct summand conjecture: any ring S which is a direct summand (as S-module) of a regular ring R is Cohen-Macaulay.

The syzygy conjecture [Evans-Griffiths '81]: any n-th syzygy module of a finite module M of projective dimension > n has rank ≥ n.

We refer to [START_REF]La conjecture du facteur direct[END_REF][3][4] [START_REF] Heitmann | Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic[END_REF] and to the earlier papers [6, ch. 9][10][15] [START_REF] Huneke | Applications of the existence of big Cohen-Macaulay algebras[END_REF] for this circle of ideas.

2.3.

The role of perfectoids in the proof. In mixed characteristic, Theorem 2.1.1 is proved using "deep ramification" techniques; more precisely, using the theory of perfectoid spaces introduced by Scholze [START_REF] Scholze | Perfectoid spaces[END_REF].

Let us review the basic notions and constructions used in the proof.

2.3.1. Perfectoid valuation rings. This is a complete, non discrete valuation ring K o of mixed characteristic (0, p), such that for some (equivalently: any)

∈ K o with p ∈ p K o , the Frobenius map F : K o / x →x p → K o / p is an isomorphism 2 . 2.3.1. Example. Let W (k) be the Witt ring of a perfect residue field k. Then K o = W (k) p 1/p ∞
is a perfectoid valuation ring (one can take = p 1/p ).

2.3.2. Perfectoid K o -algebras. These are p-adically complete, p-torsionfree K o -algebras A such that the Frobenius map F : A/

x →x p → A/ p is an isomorphism.

Using non-archimedean geometry, one can attach to them some spaces, and by glueing, construct the geometry of perfectoid spaces. This geometry will not appear explicitly in the sequel: it is hidden in the proof of the next two theorems. Nor will appear the fundamental tilting equivalence relating (perfectoid) geometry over K o and over the valuation ring K o := lim F K o / K o of characteristic p (also hidden in the proofs).

2.3.2.

Example. Let K o be as in the previous example. The p-adic completion A of

∪ i W (k)[p 1/p i ][[x 1/p i 1 , . . . , x 1/p i n ]] is a perfectoid K o -algebra.
One can construct further perfectoid algebras by adjoining p 1/p ∞ -roots of elements of A in a suitable sense: 2.3.3. Theorem. [START_REF]La conjecture du facteur direct[END_REF](improved in [12, §16.9]) The completed p-root closure3 of A[g 1/p ∞ ] is perfectoid and faithfully flat over A.

2.3.3. Almost algebra. (Faltings, Gabber-Ramero [START_REF] Gabber | Almost ring theory[END_REF]): given a commutative ring V and an idempotent ideal I, almost algebra systematically "neglects" V-modules which are killed by I (almost-zero modules). A morphism is an almost-isomorphism if its kernel and cokernel are almost-zero, etc. Whereas this is a special case of Gabriel localization, many notions of almost algebra go well beyond and are not of categorical nature: almost finite, almost etale etc.

The usual set-up in the perfectoid theory is (V, I) = (K o , p

1 p ∞ K o ).
However, for our constructions, we need to work with the following non-valuative set-up

(V, I) = (K o [t 1/p ∞ ], (pt) 1 p ∞ K o [t 1/p ∞ ]).
One obtains the notion of almost perfectoid algebra A (containing a compatible sequence of p-power roots of some g ∈ A) on considering A as a

K o [t 1/p ∞ ]-algebra (t 1/p i → g 1/p i
) and relaxing the condition on Frobenius (only required to be an almost-isomorphism). Similarly, one obtains the notion of almost CM algebra A (containing a compatible sequence of p-power roots of some g ∈ A) on interpreting the injectivity and inequality in the definition in the almost sense.

2.3.4. Perfectoid Abhyankar lemma. The classical Abhyankar lemma asserts that under appropriate assumptions (tameness...), a ramified extension can be made etale by adjoining roots of the discriminant, rather than inverting it.

There is an analog for finite ramified extensions of perfectoid algebras:

2.3.4. Theorem. [1] Let A be a perfectoid K o [t 1/p ∞ ]-algebra, such that the image g of t in A is a nonzero divisor. Let B

2.3.5.

Application ot these tools to the construction of CM algebras. Let us come back to our complete local domain S of characteristic (0, p). For simplicity, we assume that its residue field k is perfect. We want to construct a (big) CM S-algebra. By Cohen's theorem, we may view S as a finite extension of some regular ring of the form R = W (k) [[x]].

Then an S-algebra is a CM S-algebra if and only if it is faithfully flat over R.

Let g ∈ R be such that S[1/pg] finite etale over R[1/pg].
Let us take

K o = W (k) p 1/p ∞ , and
A 0 = ∪i W (k)[p 1/p i ][[x 1/p i 1 , . . . , x 1/p i n
]] be as in the above examples (A 0 is a perfectoid K o -algebra).

Let A be completed p-root closure of A 0 [g

How to get rid of "almost"?

There are two ways: 1) Hochster's modifications (which predate our work) [START_REF] Huneke | Applications of the existence of big Cohen-Macaulay algebras[END_REF].

2) Gabber's trick: replacing B by Σ -1 (B N /B (N) ), where Σ is the multiplicative system (pg) i , i → 0 ∈ N[1/p], one gets a genuine perfectoid CM S-algebra [12, §17.5].

Weak functoriality uses similar techniques, but is more difficult. Here is a strong version:

2.3.5. Theorem. [4] Any finite sequence R 0 f1 → R 1 f2 → • • • fn → R n of local homomor- phisms of complete Noetherian local domains, with R 0 of mixed characteristic, fits into a commutative diagram (2.1) R 0 f1 / / R 1 f2 / / • • • fn / / R n R + 0 f1 + / / R + 1 f2 + / / • • • fn + / / R + n C 0 / / C 1 / / • • • / / C n where R + i is the absolute integral closure of R i , C i is a perfectoid CM R i -algebra if R i is of mixed characteristic (resp. a perfect CM R i -algebra if R i is of positive characteristic). Moreover, the f +
i can be given in advance.

APPLICATIONS TO SINGULARITIES

3.1. Symbolic powers. Let S be a Noetherian ring, p a prime ideal, and V (p) the corresponding subvariety of Spec S. Symbolic powers are defined by

p (n) := (p n S p ) ∩ p.
If S is an algebra of finite type over a field, p (n) = is the ideal of functions which vanish at V (p) at order at least n (Zariski).

One has p (n) ⊃ p n , and p (n) = p n if p is generated by a regular sequence (or more generally if gr p S is a domain).

The comparison of p (n) and p m , and its extension to non-prime ideals, is a classical problem (the containment problem), which has applications in complex analysis, interpolation theory (fat points) and transcendental number theory (Waldschmidt constants).

3.1.1. Theorem. [START_REF] Ma | Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers[END_REF] Let S be an excellent regular ring of dimension d.

Then for any prime p and any n, p (dn) ⊂ p n .

This was proved by Ein-Lazarsfeld-Smith in characteristic 0, using subadditivity of the "multiplier ideal", and by Hochster in characteristic p. In mixed characteristic, Ma and Schwede use a new notion of multiplier ideal in which the complex RΓ(Y, O Y ) attached to a resolution of V (p) is replaced by a perfectoid Cohen-Macaulay algebra for S/p.

Let us explain this last point in general terms: let now S be a local domain, essentially of finite type over C, and let π : Y → Spec S be a (log-)resolution of singularities. By Grauert-Riemenschneider, R i Γ(Y, ω Y ) = 0 for i > 0, whence, by local duality:

H j m (RΓ(Y, O Y )) = 0 for j < dim S. Thus RΓ(Y, O Y ) ∈ D b (S)
appears as a "derived avatar" of a CM algebra. In mixed characteristic or in characteristic p, one can replace the missing RΓ(Y, O Y ) by a suitable a (big) Cohen-Macaulay S-algebra.

3.2. Rational singularities. Remind that S (as before) "is" a rational singularity if and only if RΓ(Y, O Y ) ∼ = S. By Grauert-Riemenschneider and local duality, any rational singularity is CM.

Question: how to check that a singularity is rational without computing a resolution? A criterion by reduction mod. p, after spreading out, has been known for some time (Hara, Smith, Mehta-Srinivas): S rational singularity ⇔ (S mod. p) F -rational singularity for all p >> 0. For a Cohen-Macaulay singularity in characteristic p, F -rationality has several equivalent definitions (e.g. the top local cohomology is a simple Frobenius module), and can be checked algorithmically (Macaulay2). But the above criterion requires all p >> 04 .

Using perfectoid Cohen-Macaulay S-algebras, Ma and Schwede have shown that it suffices to check F -rationality for some p (such that spreading out and reduction mod. p make sense), which leads to an effective criterion of rationality: 3.2.1. Theorem. [20, §8] S rational singularity ⇔ (S mod. p) F -rational singularity for some p.

This may be viewed as an application of p-adic techniques to effective complex algebraic geometry. However, perfectoid CM algebras do not appear in the statement: they are hidden in the proof that the algorithm works, in the passage from characteristic p to characteristic 0 through mixed characteristic.

There are similar results for specific classes of rational singularities (log-terminal singularities).

Rational singularities were introduced by Artin in dimension 2. In the context of their work on rational singularities in mixed characteristic, Ma and Schwede, together with Carvajal-Rosas, Polstra and Tucker, came back to the case of dimension 2 and extended Artin's theory of (Gorenstein complete) rational singularities S of dimension 2 from characteristic p to mixed characteristic (0, p > 5) (with separably closed residue field). The classification is the same:

They admit a split finite regular extension S ⊂ R. They are "Du Val singularities", i.e. of the form S = R /f R , where R is a 3-dimensional complete regular ring and f can be written in one of the forms x 2 + y 2 + z n+1 (A n )

x 2 + y 2 z + z n-1 (D n )

x 2 + y 3 + z 4 (E 6 ) x 2 + y 3 + yz 3 (E 7 ) x 2 + y 3 + z 5 (E 8 ), the label referring to the graph of a minimal resolution of the double point. One difference with the equicharacteristic case is that the label does not determine S, due to the fact that the parameter p is special, being fixed by automorphisms of S: choosing p = x, y or z may lead to different singularities with the same label.

THEORY 2 . 1 .

 21 Existence and weak functoriality. The main result is the following: 2.1.1. Theorem. [2][4] (Big) Cohen-Macaulay algebras exist, and are weakly functorial.

1 p 1 p

 11 be a finite etale A[1/pg]-algebra, and let B be the integral closure of A in B . Then B is (pt) ∞ -almost perfectoid, and for any n > 0, B/p n is (pt) ∞ -almost faithfully flat and almost finite etale over A/p n .

  1/p ∞ ]: this is a perfectoid and faithfully flat R-algebra by Theorem 2.3.3. Let B := A[1/pg] ⊗ R S, which is finite etale extension of A[1/pg]. It then follows from Theorem 2.3.4 that the integral closure B of A in B is an almost perfectoid almost CM S-algebra.

it turns out that this property is equivalent to deep ramification: Ω Ko /K o = 0 (Gabber-Ramero).

the p-root closure of a p-adic ring R: elements r of R[1/p] such that r p j ∈ R for some j > 0.

or at least, in the Smith version, for some p big enough, but with an ineffective lower bound.


Y Andr

SINGULARITIES IN MIXED CHARACTERISTIC. THE PERFECTOID APPROACH

Keywords: 1991 Mathematics Subject Classification. 13D22, 13H05, 14G20 singularities, homological conjectures, big Cohen-Macaulay algebras, perfectoid spaces. 1

Singularities in mixed characteristic. The perfectoid approoach.

1.1. Commutative Algebra took its roots in Algebraic Number Theory, Algebraic Geometry and Invariant Theory at the end of the XIXth century, in the wake of Dedekind, Kronecker and Gordan. It soon provided a unified language for the three theories as well as a consolidation of their foundations.

It may be argued that Commutative Algebra started with Hilbert's three fundamental theorems (which still bear their original names: basis theorem, syzygy theorem, Nullstellensatz), and was put on firm ground in the 30's with Krull's four cardinal concepts (locality, dimension, completion and regularity).

A main source of examples and inspiration is given by affine rings (rings of functions on affine algebraic varieties over some base field). Regular local rings (those having the property that the minimal number of generators of their maximal ideal is their Krull dimension) then generalize rings of functions around a nonsingular point of an algebraic variety. Following this viewpoint, general local rings appear as algebraic counterparts of singularities (vs. nonsingular points).

The theory was so successful that its relation to Algebraic Geometry changed: it was the latter to be founded on the former, in Zariski's and Grothendieck's time. Throughout the canonical text EGA, Commutative Algebra forms an extended, crawling "Chapter 0". 1.2. However, Commutative Algebra is more than the solfeggio of the music of Algebraic Geometry. In fact, already in the late 50's, a turn happened in the story of Commutative Algebra: the homological turn, which was impulsed by Auslander, Buchsbaum and Serre, and originated in Hilbert's syzygy's theorem. From general theory of (Noetherian) commutative rings and their ideals, Commutative Algebra became the homological theory of (Noetherian) modules.

In this new viewpoint, regular local rings are those for which any finite module has a free resolution of finite length.

By the way, let us recall that syzygies generalize the notion of presentation of a module. If a finite module M over a local ring S admits a finite free resolution 0 → S bs → S b s-1 → . . . → S b 0 → M → 0, which we assume to be minimal, the b i 's are called the Betti numbers of M (so that b 0 is the minimal number of generators, b 1 the minimal number of relations between them,...), and Im( S b i → S b i-1 ) is called the i-th syzygy of M . Describing syzygies and estimating Betti numbers is a traditional and typical task of Commutative Algebra, not necessarily related to any specific problem from Algebraic Geometry.

1.

3. An important leimotiv in Commutative Algebra is that singularities are always "finite above non-singularities". One version is Noether's theorem: every affine algebra is finite over a polynomial algebra. Another version, in the complete local case, is Cohen's theorem: every complete local domain is finite over a regular complete local ring (in fact a ring of formal power series with coefficients in a field or a discrete valuation ring).

The study of singularities thus focusses on finite extensions S of a regular (local) ring. S-modules need not have finite free resolutions; in fact, their homological study is complicated and is the subject of the homological conjectures which we will touch upon. But as we will see, it is nevertheless useful to represent the ring S as a finite extension of a regular ring R. The second domain, p-adic Hodge Theory, which started with Tate's work in the late 60s, studies p-adic representations of p-adic Galois groups, especially those which come from the cohomology of algebraic varieties. Perfectoid theory is a new development (∼ 2010) which has already profoundly transformed p-adic Hodge Theory.

In the "Chapter 0" viewpoint, Commutative Algebra stands as the provider of basic concepts, tools and theorems for Algebraic or Arithmetic Geometry and its advanced developments, such as p-adic Hodge Theory.

In the present story, we will see p-adic Hodge Theory contribute backward to some basic problems of Commutative Algebra.

THE DIRECT SUMMAND THEOREM.

Let us tell the first instance of this encounter.

Let R be a Noetherian commutative ring, and R ⊂ S be a finite extension, i.e. S is a faithful R-algebra, finitely generated as an R-module (as in Noether's or Cohen's theorem). One thus has an exact sequence of finite R-modules:

( * ) 0 → R → S → S/R → 0.
Question 1: does this exact sequence splits?

Equivalently: is R a direct summand in S? Is there an R-linear map λ :

S → R with λ(1) = 1?
This basic question arose in the late 60's (Hochster [START_REF] Hochster | Contracted ideals from integral extensions of regular rings[END_REF], Raynaud-Gruson [START_REF] Raynaud | Critères de platitude et de projectivité[END_REF]), in the framework of the homological conjectures and of descent theory (independently). An algebraic geometer may not find easy to confer it any geometric meaning; this illustrates what we wrote about "Chapter 0". But here is a related question with a more immediate algebrogeometric content: Question 2: is there an S-algebra T which is faithfully flat over R?

Equivalently: is Spec S → Spec R a covering for the fpqc topology? (It is not required that T is finitely generated, nor even noetherian).

Actually, a positive answer to Question 2 implies a positive answer to Question 1. Indeed: a) any faithfully flat map R → T is pure, i.e. universally injective. b) if a composition R → S → T is pure, so is R → S, c) a finite extension R ⊂ S is pure if and only if it splits (in the R-module sense).

Example. The sequence ( * ) splits if R is a normal Q-algebra (take for λ the trace divided out by the degree). Hence Question 1 has an easy positive answer in this case, but Question 2 is much more difficult. Counter-example. ( * ) does not split for R = Q[x, y]/(xy) and its normalization S. Besides, ( * ) does not always split for normal F p -algebras. Question 1 and therefore Question 2 have negative answers in these cases.

2.2. M. Hochster's direct summand conjecture [START_REF] Hochster | Contracted ideals from integral extensions of regular rings[END_REF] (1969) concerns Question 1, namely: DSC:

The sequence ( * ) splits if R is regular.

For instance, this may be the situation of a Noether normalization of an affine ring S (R being a polynomial ring), or a Cohen presentation of a complete local domain S (R being a ring of formal power series).

The state of the art of DSC before 2016 could be summarized as follows: a) Hochster gave (short) proofs when R contains a field, b) he reduced the problem to the unramified complete local case with perfect residue field k of char. p:

R ∼ = W (k)[[x 2 , • • • , x d ]], (x 1 = p), c) R. Heitmann [13] (2002) gave a proof in dimension d ≤ 3 1 .
Here is one of Hochster's proofs for a regular complete local domain R of char. p with perfect residue field k = R/m. Let F denote the Frobenius endomorphism of R or its finite extension S, and let n be a positive integer. Then F n m is the maximal ideal of F n R and is contained in m p n , so that ∩ (R.F n m) = 0 by Krull's theorem. On the other hand, since k is perfect, R is a finite F n R-module. Since R is a regular local ring, so is F n R, and R is a free F n R-module by Kunz's theorem. Let λ be any nonzero Rlinear form on S; rescaling, we may arrange that λ(1) = 0, hence λ(1) / ∈ R.F n m for some n, so that λ(1) is part of a basis of the F n R-module R by Nakayama's lemma. Therefore, there exists a F n R-linear form µ on 1 in dimension ≤ 2, this is easy: up to replacing S by its normalization, it can be assumed to be a reflexive R-module; it is then faithfully flat over the regular ring R of dimension ≤ 2, so that S/R is finite flat, hence projective, which ensures that ( * ) splits. The first difficult case is dimension 3. Heitmann's proof already has a flavor of "almost algebra", which will play a fundamental role in the general case.

R such that µλ(1) = 1 and in particular, µλ |F n S provides a splitting of F n R ⊂ F n S. By transport of structure via F n , ( * ) also splits.

The case of the extension R → R 1 p is already very interesting; local rings R of char. p for which its splits are called F -pure2 .

2.3.

As for Question 2, Hochster and C. Huneke have given a positive answer in the geometric case, i.e. in presence of a base field. This is much more difficult than the above short argument, and it turns out that (unlike Question 1) the char. 0 case is even more difficult (and is deduced from) the char. p case. They proved that if R is a complete regular local domain of char. p and S is a finite extension domain, then the absolute integral closure R + of R (i.e. the integral closure in an algebraic closure of the fraction field), viewed as an S-algebra, is faithfully flat over R [START_REF]Infinite integral extensions and big Cohen-Macaulay Algebras[END_REF].

2.4.

The direct summand conjecture is now a theorem:

Theorem (A. [2] 2016). Any finite extension of a regular ring splits (as a module).

In fact, what is stronger, Question 2 also admits a positive answer for regular rings: 2.4.2. Theorem. [START_REF]La conjecture du facteur direct[END_REF] For any finite extension S of a regular ring R, there is an S-algebra T which is faithfully flat over R. After Hochster's work, it suffices to deal, in both theorems, with the mixed characteristic case, and more specifically with

R = W (k)[[x 2 , • • • , x d ]]
where W (k) stands for the Witt ring of a perfect field k of char. p. In that case, and if T is p-torsion free and p-adically complete, faithful flatness over R can be checked by faithful flatness of T /p over R/p.

The simple argument sketched above for DSC in char. p does not extend to the mixed characteristic case: R has a natural Frobenius endomorphism, but it does not extend to the finite extension S in general. This suggests to introduce a mixed characteristic analog of the perfect closure, by introducing p th -power roots of the system of parameters x 1 = p, x 2 , • • • , x d , which brings us into the perfectoid world.

3. PERFECTOID NOTIONS.

3.1.

Let us begin with the notion of perfectoid field, whose origin dates back to J. Tate's studies in Galois cohomology. Let K be a complete nonarchimedean field, K o its valuation ring, and K oo its valuation ideal.

Assume that the valuation is not discrete (equivalently: K oo = (K oo ) 2 ), and that the residue field k is of char. p > 0.

3.1.1. Proposition (Gabber-Ramero [START_REF] Ramero | Almost ring theory[END_REF]). The following are equivalent:

i) K o /p x →x p → K o /p is surjective, ii) for each finite separable L/K, L o is almost etale over K o , i.e. Ω L o /K o is killed by K oo .
Such a field K is called perfectoid (if one refers to i)) or deeply ramified (if one refers to ii)). The residue field k is then a perfect field.

Example.

Let K 0 = W (k)[ 1 p ] and K be the completion of K 0 [p 1 p ∞ ]
. This is the basic perfectoid field in the sequel. Condition i) is easy to checked directly, while condition ii) tells us that K absorbs almost all the ramification of the completion K0 of the algebraic closure of K 0 (this was used by Tate to compute the Galois cohomology of K0 ).

Here and in the proposition, "almost" is used in the sense of Almost Algebra: given a commutative ring V and an idempotent ideal m, one "neglects" all V-module killed by m. Almost algebra, introduced by Faltings [START_REF] Faltings | Almost étale extensions[END_REF] and developed by O. Gabber and L. Ramero [START_REF] Ramero | Almost ring theory[END_REF], goes much beyond categorical localization, and studies (non-categorical) notions such as almost finite, almost flat, almost etale.

We shall write "p

1 p ∞ -almost" to specify that the set-up is (V, m) = (K o , K oo = p 1 p ∞ K o ): "p 1 p ∞ -
almost zero" means "killed by all fractional powers p 1 p i ".

3.2.

The generalization from fields to algebras, and further to spaces, was initiated by G. Faltings, and fully developed by P. Scholze (and also, to some extent and independently, by K. Kedlaya and R. Liu).

Let A be a Banach K-algebra, and A o be its sub-K o -algebra of powerbounded elements.

Definition (Scholze

). A is perfectoid if A o /p 1/p x →x p → A o /p is an isomorphism.
The norm of such an algebra A is equivalent to the spectral norm, and A o is the unit ball for the latter.

Example. A o = R ∞ := ∪W (k)[p 1 p i ][[x 1 p i 2 , • • • , x 1 p i d ]], A = R ∞ [ 1 p ].
A fundamental result of perfectoid theory is the so-called "almost purity theorem": 3.2.2. Theorem (Faltings; Scholze [START_REF] Scholze | Perfectoid spaces[END_REF], Kedlaya-Liu [START_REF] Kedlaya | Relative p-adic Hodge theory, I: Foundations[END_REF]). Let A be a perfectoid algebra over a perfectoid field, and B be a finite etale A-algebra.

Then B is perfectoid, and B o is an p

. S ⊗ R R ∞ [ 1 p ] may not be etale over R ∞ [ 1 p ],
hence one cannot apply Almost Purity: a non-etale finite extension of a perfectoid algebra need not be perfectoid.

To remedy this, we take inspiration from Abhyankar's classical lemma, which tells that under appropriate assumptions, one can achieve etaleness by adjoining roots of the discriminant.

We follow this strategy. Let

g ∈ R = W (k)[[x 2 , • • • , x d ]] be a discrim- inant of S[ 1 p ]/R[ 1 p ]
. The first step is to note that adjoining p th -power roots of g, in the (non-naive

) sense of considering R ∞ [ 1 p ] g 1 p ∞ o
, is "harmless", namely:

4.1.1. Theorem (A. [START_REF]La conjecture du facteur direct[END_REF]). Let A be a perfectoid K-algebra, and let g ∈ A o be a non-zero divisor. Then for any n, A g

1 p ∞ o /p m is p 1 p ∞ -almost faithfully flat over A o /p m . In particular, R ∞,g := R ∞ [ 1 p ] g 1 p ∞ o has the property that R ∞,g /p is p 1 p ∞ - almost faithfully flat over R ∞ /p. Note that R ∞,g is in general much bigger (even before completion) than the ring R ∞ [g 1 p ∞ ]: it contains elements such as p -1 p (g σ -1 -g 1 p ), where σ : W (k)[[x 1 p 2 , • • • , x 1 p d ]] → W (k)[[x 2 , • • • , x d ]] is the natural W (k)-semilinear Frobenius isomorphism.
The proof of the theorem uses a deformation argument: one spreads out the perfectoid space attached to A o ∞ into a perfectoid space Y by adding one variable x. Let Y < be the (perfectoid) -tubular neighborhood of the locus x = g. Then:

• A g 1 p ∞ o is p 1 p ∞ -almost equal to colim O + (Y < ). • Using Scholze's (almost) description of O + (Y < )/p in terms of "Puiseux-like" series in the variable x -g with coefficients in O + (Y )/p , one shows that O + (Y < )/p is almost faithfully flat over O + (Y )/p .
For details, we refer to [START_REF]La conjecture du facteur direct[END_REF] (and/or to the short account [START_REF]Perfectoid spaces and the homological conjectures[END_REF]) 3 .

3 this theorem and its avatars also play an important role in the latest development of perfectoid theory: prism and prismatic cohomology (Bhatt, Scholze).

The next step represents at the same time a perfectoid version of Abhyankar's lemma, and a ramified version of the Almost Purity theorem.

4.2.1. Theorem (A. [START_REF] André | Le lemme d'Abhyankar perfectoïde[END_REF]). Let A be a perfectoid K-algebra, containing a sequence of p th -power roots of a non-zero divisor g ∈ A o (Ex:

A = R ∞,g [ 1 p ]). Let B be a finite etale A[ 1 g ]-algebra, and B o be the integral closure of A o in B (hence B o [ 1 pg ] = B ). Then for any n, B o /p n is an (pg) 1 p ∞ -almost finite etale A o /p n algebra.
The proof also uses a deformation argument. Let X be the perfectoid space attached to A o = O + (X), and X > be the (perfectoid) complement of -tubular neighborhood of the discriminant locus g = 0. Then:

• O + (X) is (pg) 1 p ∞ -almost equal to lim O + (X > ) (

by Scholze's perfectoid Riemann extension theorem).

• By almost purity over

X > , (B O(X > )) + is almost finite etale over O + (X > ).
The bulk of the work is the passage to the limit → 0, for which we refer to [START_REF] André | Le lemme d'Abhyankar perfectoïde[END_REF] (and/or again to the short account [START_REF]Perfectoid spaces and the homological conjectures[END_REF]).

Let us go back to DSC, in the mixed characteristic setting

: R = W (k)[[x 2 , • • • , x d ]]
, S finite extension, etale outside pg = 0.

Applying the perfectoid Abhyankar lemma with

A = R ∞,g [ 1 p ], B = S ⊗ R A[ 1 g ],
one gets an S-algebra B o sitting on top of a tower

R α → R ∞ β → R ∞,g γ → B o
where α is faithfully flat, β is p How to get rid of "almost"? For this, we will use some "Cohen-Macaulay notions".

BIG COHEN-MACAULAY ALGEBRAS

5.1.

Let S be a Noetherian local ring, and T be a (possibly big, i.e. non-Noetherian) extension.

Definition.

T is a (big) Cohen-Macaulay S-algebra if any sequence of parameters x 1 , . . . , x d of S becomes regular in T (i.e. x 1 is non-zerodivisor in T , x 2 is non-zero divisor in T /x 1 T ..., and T = (x 1 , . . . , x d )T ).

Hochster conjectured that such a T always exists. Example. If S a complete local domain of char. p, its absolute integral closure S + is a big Cohen-Macaulay S-algebra (Hochster-Huneke, cf. 2.3).

The relevance of Cohen-Macaulay algebras to our purpose comes from the following (classical) lemma: 5.1.2. Lemma. Assume S is a complete local domain, hence a finite extension of complete regular local ring R .

Then T is a (big) Cohen-Macaulay S-algebra if and only if T is Rfaithfully flat.

Flatness of Cohen-Macaulay algebras T over a regular local ring R can be seen as follows. Any finite R-module M has projective dimension ≤ d = dim R, hence Tor R i (M, T ) = 0 for all i > d. One argues by descending induction on i. By dévissage, one may assume that M = R/p for a prime p. If (x 1 , . . . , x i ) is a maximal regular sequence contained in p, M embeds into

N := R/R(x 1 , . . . , x i ). But Tor R i (N, T ) = 0 for all i > 0 if T is a Cohen-Macaulay R-algebra. The exact sequence Tor R i+1 (N/M, T ) → Tor R i (M, T ) → Tor R i (N, T ) is thus the 0-sequence, hence T is a flat R- module.

The problem to pass from an almost

Cohen-Macaulay S-algebra T - (such as B o at the end of the previous section) to a genuine Cohen-Macaulay S-algebra T can be settled by Hochster's (classical) technique of algebra modifications [START_REF]Big Cohen-Macaulay algebras in dimension three via Heitmann's theorem[END_REF] [START_REF]-Applications of the existence of big Cohen-Macaulay algebras[END_REF]. The idea is to force each relation x i+1 t i+1 = i 1 x j t j with t j ∈ T -, x i ∈ S, to come from a relation t i+1 = i 1 x j t j by introducing step by step new variables: a partial modification of degree n of an S-module M is a homomorphism M → M where, given a relation

x i+1 m i+1 = i 1 x j m j with coefficients m j in M , M := M [T 1 , . . . , T i ] ≤n /(m i+1 -i 1 x j T j ) • M [T 1 , . . . , T i ] ≤n-1 .
Starting from T -and taking the colimit of partial modifications for all such relations yields an S-algebra T for which x i+1 is not a zero divisor mod. x 1 , . . . , x i . Condition T = (x 1 , . . . , x d )T comes from the assumption that T -is almost Cohen-Macaulay.

This provides the last step in the proof of Hochster's conjecture about the existence of big Cohen-Macaulay algebras. Using the above lemma, this implies Theorem 2.4.2 and the Direct Summand Conjecture 2.4.1 (positive answer to Question 1 and 2 for regular rings).

5.

3. This is the first item of the following 5.3.1. Theorem (A. [START_REF]La conjecture du facteur direct[END_REF], [START_REF]Weak functoriality of Cohen-Macaulay algebras[END_REF]).

(1) Any Noetherian local ring S has a big Cohen-Macaulay algebra T .

Y. ANDR É

(2) For any local morphism S → S of Noetherian complete local domains, there is a morphism of respective big Cohen-Macaulay algebras T → T .

The second item is the so-called "weak functoriality of big Cohen-Macaulay algebras", also conjectured by Hochster. Its poof is more subtle and requires a delicate consideration of integral perfectoid algebras (the most difficult case is when S is of mixed characteristic and S of char. p).

In mixed characteristic, one has a more precise result regarding item (1) (which is used in the proof of item (2)): [START_REF]Integral perfectoid big Cohen-Macaulay algebras via André's theorem[END_REF], A. [START_REF]Weak functoriality of Cohen-Macaulay algebras[END_REF]). Any Noetherian complete local domain S of char. (0, p) admits an (integral) perfectoid 4 Cohen-Macaulay algebra T . K. Shimomoto's idea is to use the tilting equivalence between perfectoid algebras over K and certain perfect algebras over a perfect field

5.3.2. Theorem (Shimomoto
K ∼ = k((t 1 p ∞
)) of char. p (Scholze): one can apply Hochster modifications in char. p, after tilting, and then untilt. An alternative and more recent argument, due to Gabber [START_REF] Gabber | Some remarks on big Cohen-Macaulay algebras[END_REF], uses ultraproducts instead of tilting and Hochster modifications.

5.4. Kunz' theorem in mixed characteristic. In the same vein but independently from the above results, B. Bhatt, S. Iyengar and L. Ma extended Kunz's flatness criterion in char. p to mixed characteristic. Let us recall that for a Noetherian ring R of char. p, Kunz' theorem asserts that R is regular if and only if R

x →x p → R is flat (which amounts to saying that there exists a perfect, faithfully flat R-algebra 5 ).

Let R now be any Noetherian p-adically complete ring. 5.4.1. Theorem (Bhatt-Iyengar-Ma [START_REF] Iyengar | Regular rings and perfect(oid) algebras[END_REF] 2018). R is regular if and only if there exists an integral perfectoid 6 , faithfully flat, R-algebra.

APPLICATIONS TO THE HOMOLOGICAL CONJECTURES

6.1. The homological turn. We have already mentioned the homological turn in commutative algebra in the 60's: from the study of Noetherian rings and their ideals (Krull, Zariski...) to the study of the homological properties of their modules (Auslander, Buchsbaum, Serre...). 4 an integral perfectoid K o -algebra is a p-adically complete p-torsionfree K o -algebra T such that the Frobenius map T /p 1 p → T /p is an isomorphism. 5 such an algebra is a (big) Cohen-Macaulay R-algebra. 6 in some generalized sense, which does not require a perfectoid base field.

Example. A local ring R is regular if and only if every finite R-module has a finite free resolution.

C. Peskine and L. Szpiro introduced reduction techniques to char. p, and the crucial observation that in char. p, Frobenius preserves finite free resolutions (which may be seen as an extension of Kunz' theorem).

Example. A local ring S is Cohen-Macaulay if and only if there is an S-module of finite length with a finite free resolution [START_REF] Peskine | Dimension projective finie et cohomologie locale[END_REF], [START_REF] Roberts | Le théorème d'intersection[END_REF].

We refer to [7][18] for accounts of the homological conjectures (before the introduction of perfectoid techniques). The relevance of Cohen-Macaulay algebras typically comes from the following observation: let

(F * ) 0 → F s → F s-1 → . . . → F i φ i → F i-1 → . . . F 1 → F 0
be a complex of finite free S-modules, let r i be the rank of Im φ i , and let I r i (φ i ) be the Fitting ideal of S generated the minors of size r i of φ i .

If (F * ) is acyclic, then codim I r i (φ i ) ≥ i (Buchsbaum-Eisenbud). The converse is not true in general if S is not Cohen-Macaulay; however if codim I r i (φ i ) ≥ i, then (F * ) ⊗ S T is acyclic for any Cohen-Macaulay Salgebra T (cf. [7, 9.1.8]).

6.2. Pure subrings of regular rings. The following consequence of weak functoriality of Cohen-Macaulay algebras generalizes and extends in mixed characteristic the well-known theorem of Hochster-Roberts about Cohen-Macaulayness of rings of invariants of polynomials under reductive groups in char. 0. 6.2.1. Theorem (Heitmann-Ma [START_REF] Heitmann | Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic[END_REF]; A. [START_REF]Weak functoriality of Cohen-Macaulay algebras[END_REF]). Any pure subring of a regular ring (e.g. any subring which is a direct summand as a module) is Cohen-Macaulay.

Let us recall the (classical) derivation of the theorem from 5.3.1 [START_REF]La conjecture du facteur direct[END_REF]. We denote the embedding by S → R , where R is regular. We may also assume that S is a finite extension of a regular ring R, and it suffices to show that S is flat over R. Let T S → T R be a compatible map of Cohen-Macaulay algebras. For any R-module M , one has a commutative square In fact, Heitmann and Ma proved just the right amount of weak functoriality needed to get the corollary, but they also obtained more: S has pseudo-rational singularities.

Tor R i (M, S) a ---→ Tor R i (M, R )   b   Tor R i (M, T S ) ---→ Tor R i (M, T R ).
6.3. The syzygy theorem. Let us turn to the traditional problem of estimating Betti numbers and ranks of syzygies of noetherian modules.

Let S be a Noetherian local ring, and let M be a finite S-module which admits a finite free resolution. A fortiori, b i ≥ 2i + 1 if i < s -1, and b s-1 ≥ s.

These bounds are optimal in general; for instance, if S is the localization of k[x 1 , x 2 , x 3 ] at 0 and M = k, the minimal free resolution is 0 → S → S 3 → S 3 → S → k → 0, so that s = 3 and the first two syzygies have rank 1 and 2 respectively. E. Evans and P. Griffiths [START_REF] Evans | The syzygy problem[END_REF] stated and proved this theorem when S contains a field, using the existence of Cohen-Macaulay algebras in this case. Hochster [START_REF]Canonical elements in local cohomology modules and the direct summand conjecture[END_REF] and T. Ogoma [START_REF] Ogoma | A note on the syzygy problem[END_REF] proved that the statement follows from DSC in general. DSC being proven (thanks to perfectoid techniques), these bounds are now unconditional.

Similar bounds hold for minimal injective resolutions, S being replaced by the injective hull of the residue field (cf. [7, 9.6]).

Actually, most of the homological conjectures which were standing for a while on Hochster's list are now solved, as consequences of theorem 5.3.1 (still, a number of related old problems remain open, such as Serre's positivity conjecture for intersection numbers and several questions posed by Peskine and Szpiro in their thesis).

APPLICATIONS TO SINGULARITIES

7.1. Starting from theorems 5.3.1 and 5.3.2, Ma and K. Schwede [START_REF] Ma | Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers[END_REF][25] developed an analog of tight closure theory in mixed characteristic, with applications to singularities. In their theory, integral perfectoid Cohen-Macaulay algebras play somehow the role of resolution of singularities in char. 0.

Let S be a local domain, essentially of finite type over C, and let π : Y → Spec S be a resolution of singularities. By Grauert-Riemenschneider, R i Γ(Y, ω Y ) = 0 for i > 0, whence H j m (RΓ(Y, O Y )) = 0 for j < dim S by local duality. Thus RΓ(Y, O Y ) appears as a "derived avatar" of a Cohen-Macaulay algebra. Since in many questions about singularities, it is not so much resolution itself which matters but the object RΓ(Y, O Y ) ∈ D b (S), the idea is to replace, in mixed characteristic or in char. p, this object by suitable (big) Cohen-Macaulay S-algebras.

7.2.

Let us just outline one striking application of this idea to rational singularities. Recall that, by definition, S (as before) "is" a rational singularity if and only if RΓ(Y, O Y ) ∼ = S. In particular, by Grauert-Riemenschneider and local duality, S is Cohen-Macaulay.

It has been known for a long time (N. Hara [START_REF] Hara | A characterization of rational singularities in term of injectivity of Frobenius maps[END_REF], K. Smith [START_REF] Smith | F-rational rings have rational singularities[END_REF], V. Mehta and V. Srinivas [START_REF] Mehta | A characterization of rational singularities[END_REF]) that S is a rational singularity if and only if, after "spreading out", (S mod. p) is a F -rational singularity (i.e. it is Cohen-Macaulay and in addition, the top local cohomology module is a simple Frobenius module) for p >> 0. While F -rationality is a checkable property (Macaulay2), checking it for all p >> 0 does not give rise to an algorithm.

But remarkably, Ma and Schwede recently proved [START_REF]Singularities in mixed characteristic via big Cohen-Macaulay algebras[END_REF] that it suffices to check that (S mod. p) is a F -rational singularity for some p. This provides an efficient algorithm: in practice, one may choose a very small prime p.

The proof goes through mixed characteristic. More precisely, it uses a perfectoid avatar of the notion of rational singularity in mixed characteristic as link between char. p and char. 0. In some sense, this is an application of perfectoid theory to complex geometry! A similar result holds for log-terminal singularities (rational singularities for which the multiplier ideal is trivial); in this case, the corresponding property in char. p has a simple formulation: for any non zero divisor s ∈ S, there is a positive integer n such that S •s 1 p n → S 1 p n splits 7 . These results may be the beginning of a program toward making the connection between singularity theory and birational geometry in char. 0 and their counterparts in char. p both effective and direct (even though the proof that the algorithms work would be indirect and go through mixed characteristic and perfectoid techniques).

7.3.

Another striking application concerns symbolic powers. For any Noetherian ring S, prime ideal p, and positive integer n, the symbolic power is defined by p (n) := (p n S p ) ∩ p.

In the affine case, this is just the ideal of functions which vanish at V (p) at order at least n. Obviously, p (n) ⊃ p n , and if p is generated by a regular sequence, it is well-known that p (n) = p n . To compare p (n) and p n in general 7 for the intermediate notion of log-canonical singularity, the situation is more subtle: for instance, the origin of the affine Fermat surface of degree 3 is a log-canonical singularity, but (S mod. p) is F -pure if and only if p ≡ 1(3). See also [START_REF] Ishii | Introduction to singularities[END_REF]Conj. 10.3.15] and [START_REF] Takagi | F-singularities[END_REF].

is a classical problem which has far-reaching applications, e.g. in complex analysis or in transcendental number theory (Waldschmidt constants). 7.3.1. Theorem (Ma, Schwede [START_REF] Ma | Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers[END_REF]). Let S be excellent regular of dimension d. Then for any prime p and any positive integer n, p (dn) ⊂ p n .

In char. 0, this was proven by Ein-Lazarsfeld-Smith using subadditivity of the multiplier ideal. In mixed characteristic, Ma and Schwede follow the same strategy, defining a new notion of multiplier ideal in which the complex RΓ(Y, O Y ) attached to a log-resolution of V (p) is replaced by an integral perfectoid Cohen-Macaulay algebra for S p .

1. 4 .

 4 We shall report on recent progresses which came from an encounter between two domains:Commutative Algebra -------p-adic Hodge Theory (Hilbert, Krull, ...) (Tate, Fontaine, Faltings...) Noetherian world non-Noetherian world (finite-dimensional rings, (non-archimedean finite modules) Banach algebras...)Homological conjecturesnew ← ---perfectoid theory (Peskine-Szpiro, Hochster...) (Scholze, ...)

1 p 4 . 4 . 1 .

 1441 ∞ -almost finite etale A o algebra. PERFECTOID ABHYANKAR LEMMA Let us come back to the situation of DSC, and keep the above notation

1 p

 1 ∞ -almost faithfully flat mod. p, γ is (pg) 1 p ∞ -almost faithfully flat mod. p. It follows that B o /p is almost isomorphic to a faithfully flat R/p-algebra. Thus B o is "almost" our wanted T (cf. 2.4).

  in which a, b are injective since S → R → T R are pure morphisms (see e.g. [2, 5.1.1]), and Tor R i (M, T S ) = 0 for i > 0 since T S is Cohen-Macaulay over S, hence flat over R; therefore Tor R i (M, S) = 0.

6. 3 . 1 .

 31 Theorem. Let 0 → S bs → S b s-1 → . . . → S b 0 → M → 0 be a minimal free resolution of M . Then the i-th syzygy has rank ≥ i for any i ≤ s -1.

beyond the regular case, they contain char. p analogs of log-canonical singularities.