A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras
aa r X i v : . [ m a t h . A C ] N ov A VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOSTCOHEN-MACAULAY ALGEBRAS
KEI NAKAZATO AND KAZUMA SHIMOMOTO
Abstract.
In this paper, we prove that a complete Noetherian local domain of mixed characteristic p > p . This result is seen as amixed characteristic analogue of the fact that the perfect closure of a complete local domain inpositive characteristic is almost Cohen-Macaulay. To this aim, we carry out a detailed study ofdecompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of Andr´e’sperfectoid Abhyankar’s lemma and Riemann’s extension theorem. Contents
1. Introduction 11.1. Main results on the decompletion of perfectoids 21.2. Main results on commutative algebra 42. Notation and conventions 53. Preliminary lemmas 73.1. Complete integral closure under completion 74. A variant of perfectoid Abhyankar’s lemma for almost Witt-perfect rings 104.1. Variants of Riemann’s extension theorems 124.2. Witt-perfect Abhyankar’s lemma 215. Applications of Witt-perfectoid Abhyankar’s lemma 265.1. Construction of almost Cohen-Macaulay algebras 265.2. Construction of big Cohen-Macaulay modules 316. Appendix: Integrality and almost integrality 32References 331.
Introduction
In the present article, rings are assumed to be commutative with a unity. Recently, Yves Andr´eestablished
Perfectoid Abhyankar’s Lemma in [1] as a conceptual generalization of
Almost PurityTheorem ; see [50, Theorem 7.9]. This result is stated for perfectoid algebras over a perfectoid field,which are defined to be certain p -adically complete and separated rings. Using his results, Andr´eproved the existence of big Cohen-Macaulay algebras in mixed characteristic in [2]. More precisely,he constructed a certain almost Cohen-Macaulay algebra using perfectoids. We are inspired bythis result and led to the following commutative algebra question, which is raised in [47] and [48]implicitly.
Key words and phrases.
Almost purity, big Cohen-Macaulay algebra, ´etale extension, perfectoid space.2010
Mathematics Subject Classification : 13A18, 13A35, 13B05, 13D22, 13J10, 14G22, 14G45.
Question 1 (Roberts) . Let ( R, m ) be a complete Noetherian local domain of arbitrary characteristicwith its absolute integral closure R + . Then does there exist an R -algebra B such that B is an almostCohen-Macaulay R -algebra and R ⊂ B ⊂ R + ? Essentially, Question 1 asks for a possibility to find a relatively small almost Cohen-Macaulayalgebra. The structure of this article is twofold. We begin with its arithmetic side and move to itsapplication to Question 1.1.1.
Main results on the decompletion of perfectoids.
The first goal is to relax the p -adiccompleteness from Perfectoid Abhyankar’s Lemma and incorporate the so-called Witt-perfect con-dition, which is introduced by Davis and Kedlaya in [16]. Roughly speaking, a Witt-perfect (or p -Witt-perfect) algebra is a p -torsion free ring A whose p -adic completion becomes an integralperfectoid ring. Indeed, Davis and Kedlaya succeeded in proving the almost purity theorem forWitt-perfect rings. The present article is a sequel to authors’ previous work [44], in which theauthors were able to give a conceptual proof to the almost purity theorem by Davis-Kedlaya byanalyzing the integral structure of Tate rings under completion. The advantage of working withWitt-perfect rings is that it allows one to take an infinite integral extension over a certain p -adically complete ring to construct an almost Cohen-Macaulay algebra. The resulting algebra isnot p -adically complete, but its p -adic completion is integral perfectoid; see Main Theorem 4 below.To establish this result, let us state our first main result; see Theorem 4.19. Main Theorem 1.
Let A be a p -torsion free ring that is flat over a Witt-perfect valuation domain V of rank admitting a compatible system of p -power roots p pn ∈ V , together with a nonzerodivisor g ∈ A admitting a compatible system of p -power roots g pn ∈ A . Suppose that(1) A is p -adically Zariskian and A is completely integrally closed in A [ pg ] ;(2) A is a ( pg ) p ∞ -almost Witt-perfect ring that is integral over a Noetherian ring;(3) ( p, g ) is a regular sequence on A (or more generally, it suffices to assume that p, g arenonzero divisors on the p -adic completion b A );(4) A [ pg ] ֒ → B ′ is a finite ´etale extension.Let us put g − p ∞ A := n a ∈ A [ 1 g ] (cid:12)(cid:12)(cid:12) g pn a ∈ A, ∀ n > o , which is an A -subalgebra of A [ g ] . Denote by B := ( g − p ∞ A ) + B ′ the integral closure of g − p ∞ A in B ′ (which is equal to the integral closure of A in B ′ by Lemma 4.18). Then the followingstatements hold:(a) The Frobenius endomorphism F rob : B/ ( p ) → B/ ( p ) is ( pg ) p ∞ -almost surjective andit induces an injection B/ ( p p ) ֒ → B/ ( p ) .(b) Suppose that A is a normal ring that is torsion free and integral over a Noetheriannormal domain. Then the induced map A/ ( p m ) → B/ ( p m ) is ( pg ) p ∞ -almost finite´etale for all m > . In the original version of Perfectoid Abhyankar’s Lemma as proved in [1] and [2], it is assumedthat A is an integral perfectoid ring , which is necessarily p -adically complete and separated. Aswe will see in the course of the proof, a comparison of the valuations between A and its p -adiccompletion b A plays an essential role and hence, A is required to possess integral closedness condition VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 3 in A [ p ]. Recall that an integrally closed domain A is the intersection of all valuation domains thatlie between A and the field of fractions; see [58, Proposition 6.8.14] for the proof of this result fromclassical valuation theory. We need a variant of this result for affinoid Tate rings. The regularityassumption on the sequence ( p, g ) ensures that p, g are nonzero divisors on the p -adic completion b A ; this is due to Lemma 4.5. A detailed study of almost Witt-perfect rings appears in the paper[44]; see also Definition 4.2 below. The functor A g − p ∞ A is called a functor of almost elements ,which is fundamental in almost ring theory. The idea of the proof of Main Theorem 1 is to transportAndr´e’s original proof to our situation with the following ingredients: • The almost purity theorem for Witt-perfect rings. • Descent to Galois extensions of commutative rings. • Riemann’s extension theorem (Hebbarkeitssatz). • Comparison of continuous valuations for affinoid Tate rings under completion. • Comparison of integral closure and complete integral closure.The almost purity theorem for Witt-perfect rings is attributed to Davis and Kedlaya; see [16] and[17]. A systematic approach to this important result was carried out in authors’ paper [44]. Theassumption that A is integral over a Noetherian ring as appearing in Main Theorem 1 is required bythe following Witt-perfect Riemann’s Extension Theorem ; see Theorem 4.15 as well as notation. Itsperfectoid version has been proved by Scholze in [51], and Andr´e used it in the proof of PerfectoidAbhyankar’s Lemma in [1].
Main Theorem 2.
Let A be a p -torsion free ring that is flat over a Witt-perfect valuation domain V of rank admitting a compatible system of p -power roots p pn ∈ V , together with a nonzerodivisor g ∈ A admitting a compatible system of p -power roots g pn ∈ A . Suppose that(1) A is p -adically Zariskian and A is completely integrally closed in A [ p ] ;(2) A is a ( pg ) p ∞ -almost Witt-perfect ring that is integral over a Noetherian ring;(3) ( p, g ) is a regular sequence on A (or more generally, it suffices to assume that p, g arenonzero divisors on the p -adic completion b A ).Let us put e A := lim ←− j> A j ◦ . Then the following statements hold.(a) There is a ( pg ) p ∞ -almost isomorphism for each j > : A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) ≈ A j ◦ . (b) The natural map A j ◦ → A j ◦ induces an isomorphism: d A j ◦ ∼ = −→ A j ◦ . In particular, A j ◦ is Witt-perfect.(c) There are ring isomorphisms: A + A [ pg ] ∼ = A ∗ A [ pg ] ∼ = e A , where A + A [ pg ] (resp. A ∗ A [ pg ] ) isthe integral closure (resp. complete integral closure) of A in A [ pg ] . If moreover A iscompletely integrally closed in A [ pg ] , then A ∼ = g − p ∞ A ∼ = e A . K.NAKAZATO AND K.SHIMOMOTO
The authors do not know if the Noetherian assumption can be dropped from Main Theorem2, which is necessary in order to find valuation rings of rank 1. The proof of Main Theorem 2 isreduced to the Riemann’s extension theorem for perfectoid algebras up to p -adic completion. Asto the statement ( c ), we succeeded in giving an alternative proof, which is stated as follows; seeTheorem 4.14. Main Theorem 3.
Let A be a ring with a nonzero divisor ̟ that is ̟ -adically Zariskian andintegral over a Noetherian ring. Let g ∈ A be a nonzero divisor. Let A j the Tate ring associated to (cid:0) A [ ̟ j g ] , ( ̟ ) (cid:1) for every integer j > . Then we have an isomorphism of rings A + A [ ̟g ] ∼ = −→ lim ←− j A j ◦ , where the transition map A j +1 ◦ → A j ◦ is the natural one. In the above theorem, the assumption that A is ̟ -adically Zariskian is necessary and it is inter-esting to know if it is possible to get rid of it.1.2. Main results on commutative algebra.
Andr´e proved that any complete Noetherian localdomain maps to a big Cohen-Macaulay algebra and using his result, it was proved that such a bigCohen-Macaulay algebra could be refined to be an integral perfectoid big Cohen-Macaulay algebrain [55]. We refer the reader to Definition 5.1 and Definition 5.2 for big (almost) Cohen-Macaulayalgebras. Question 1 was stated in a characteristic-free manner. Let us point out that if dim R ≤ R + is a big Cohen-Macaulay algebra in an arbitrary characteristic. This is easily seen byusing Serre’s normality criterion. Recall that if R has prime characteristic p >
0, then R + is abig Cohen-Macaulay R -algebra. This result was proved by Hochster and Huneke and their proof isquite involved; see [30], [31], [32], [36], [46] and [49] for related results as well as [28], [29] and [40] forapplications to tight closure, multiplier/test ideals and singularities on algebraic varieties. Thereis another important work on the purity of Brauer groups using perfectoids; see [15]. It seems tobe an open question whether R + is almost Cohen-Macaulay when R has equal-characteristic zero.If R has mixed characteristic of dimension 3, Heitmann proved that R + is a ( p ) p ∞ -almost Cohen-Macaulay R -algebra in [26]. Our main concern, inspired also by the recent result of Heitmann andMa [29], is to extend Heitmann’s result to the higher dimensional case, thus giving a positive answerto Roberts’ question in mixed characteristic as an application of Main Theorem 1; see Theorem5.5. Main Theorem 4.
Let ( R, m ) be a complete Noetherian local domain of mixed characteristic p > with perfect residue field k . Let p, x , . . . , x d be a system of parameters and let R + be the absoluteintegral closure of R . Then there exists an R -algebra T together with a nonzero element g ∈ R suchthat the following hold:(1) T admits compatible systems of p -power roots p pn , g pn ∈ T for all n > .(2) The Frobenius endomorphism F rob : T / ( p ) → T / ( p ) is surjective.(3) T is a ( pg ) p ∞ -almost Cohen-Macaulay normal domain with respect to p, x , . . . , x d and R ⊂ T ⊂ R + .(4) R [ pg ] → T [ pg ] is an ind-´etale extension. In other words, T [ pg ] is a filtered colimit of finite´etale R [ pg ] -algebras contained in T [ pg ] . VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 5
In other words, one can find an almost Cohen-Macaulay, Witt-perfect normal domain (its p -adiccompletion is integral perfectoid) between R and its absolute integral closure. Using Hochster’s par-tial algebra modification and tilting, one can construct an integral perfectoid big Cohen-Macaulay R -algebra over T ; see [55] for details. In a sense, Main Theorem 4 is regarded as a weak analogue ofthe mixed characteristic version of a result by Hochster and Huneke. The proof of our result doesnot seem to come by merely considering decompleted versions of the construction by Heitmann andMa in [29], due to the difficulty of studying ( pg ) p ∞ -almost mathematics under completion. Thisis the main reason one is required to redo the decompletion of Andr´e’s results in [1] and [2]. Thisarticle is also intended to provide essential ideas surrounding the original proof of the perfectoidAbhyankar’s lemma. Finally, Bhatt recently proved that the absolute integral closure of a com-plete local domain ( R, m ) of mixed characteristic has the property that R + /p n R + is a balancedbig Cohen-Macaulay R/p n R -algebra for any n >
0; see [10]. It will be interesting to know how hismethods and results are compared to ours; at present, the authors have no clue. However, it isworth pointing out the following fact. • The almost Cohen-Macaulay algebra T constructed in Main Theorem 4 is integral over theNoetherian local domain ( R, m ) and much smaller than the absolute integral closure R + .In a sense, T is a ”smallest” almost Cohen-Macaulay algebra over R . We mention some potentialapplications of Main Theorem 4.(1) Connections with the singularities studied in [41] by exploiting the ind-´etaleness of R [ pg ] → T [ pg ].(2) A refined study of the main results on the closure operations in mixed characteristic asdeveloped in [37].(3) An explicit construction of a big Cohen-Macaulay module from the R -algebra T ; see Corol-lary 5.9. Caution : In this paper, we take both integral closure and complete integral closure for a givenring extension. This distinction is not essential in our setting in view of Proposition 6.1. However,we opt to formulate the results (mostly) in complete integral closure, because we believe that correctstatements of the possible generalizations of our main results without integrality over a Noetherianring should be given in terms of complete integral closure. The reader is warned that completeintegral closure is coined as total integral closure in the lecture notes [8].2.
Notation and conventions
We use language of almost ring theory. The most exhaustive references are [21] and [22], wherethe latter discusses applications of almost ring theory to algebraic geometry and commutative ringtheory. Notably, it includes an extension of the Direct Summand Conjecture to the setting oflog-regular rings. Throughout this article, for an integral domain A , let Frac( A ) denote the fieldof fractions of A . A basic setup is a pair ( A, I ), where A is a ring and I is its ideal such that I = I . Moreover, we usually assume that I is a flat A -module. An A -module M is I-almost zero (or simply almost zero ) if IM = 0. Let f : M → N be an A -module map. Then we say that f is I-almost isomorphic (or simply almost isomorphic ) if kernel and cokernel of f are annihilated by I . Let us define an important class of basic setup ( K, I ) as follows: Let K be a perfectoid field ofcharacteristic 0 with a non-archimedean norm | · | : K → R ≥ . Fix an element π ∈ K such that | p | ≤ | π | < I := S n> π pn K ◦ (such an element π exists and plays a fundamental role inperfectoid geometry). Set K ◦ := { x ∈ K | | x | ≤ } and K ◦◦ := { x ∈ K | | x | < } . Then K ◦ is acomplete valuation domain of rank 1 with field of fractions K and the pair ( K ◦ , I ) is a basic setup. K.NAKAZATO AND K.SHIMOMOTO
Let (
A, I ) be a basic setup. Then the category of almost A -modules or A a -modules A a − Mod ,is the quotient category of A -modules A − Mod by the Serre subcategory of I -almost zero modules.So this defines the localization functor ( ) a : A − Mod → A a − Mod . This functor admits a rightadjoint and a left adjoint functors respectively:( ) ∗ : A a − Mod → A − Mod and ( ) ! : A a − Mod → A − Mod . These are defined by M ∗ := Hom A ( I, M ) with M a = M and M ! := I ⊗ A M ∗ . See [21, Proposition2.2.14 and Proposition 2.2.23] for these functors. So we have the following fact: The functor ( ) ∗ commutes with limits and ( ) ! commutes with colimits. Finally, the functor ( ) a commutes withboth colimits and limits. In particular, an explicit description of M ∗ will be helpful. Henceforth,we abusively write M ∗ for ( M a ) ∗ for an A -module M . The notation M ≈ N will be used throughout to indicate that M is I -almost isomorphic to N . We also say that an A -module map M → N is I -almost isomorphic. Lemma 2.1.
Let M be a module over a ring A and let t, π ∈ A be nonzero divisors on both A and M such that A admits a compatible system of p -power roots π pn ∈ A for n ≥ . Set I = S n> π n A with π n := π pn . Then ( A, I ) is a basic setup and there is an equality: M ∗ = n b ∈ M [ 1 π ] (cid:12)(cid:12)(cid:12) π n b ∈ M for all n > o . Moreover, the natural map M → M ∗ is an I -almost isomorphism. If M is an A -algebra, then M ∗ has an A -algebra structure. Finally, M is t -adically complete if and only if M ∗ is t -adicallycomplete.Proof. The presentation for M ∗ is found in [50, Lemma 5.3] over a perfectoid field and the proofthere works under our setting without any modifications. If M is an A -algebra, then the abovepresentation will endow M ∗ with an A -algebra structure. In other words, M ∗ is naturally an A -subring of M [ π ]. Finally, ( ) ∗ commutes with limits and t is a nonzero divisor of M , the proof of[50, Lemma 5.3] applies to conclude that M ∗ is t -adically complete if and only if M is t -adicllaycomplete. Another proof of this lemma is found in [8, Proposition 4.4.3] (cid:3) In the situation of the lemma, we often write M ∗ as T n> π − pn M or π − p ∞ M to indicate thatwhat basic setup of almost ring theory we are talking. We need some basic language from Huber’scontinuous valuations and adic spectra; see [33] and [34]. Definition 2.2.
A topological ring A is called Tate , if there is an open subring A ⊂ A togetherwith an element t ∈ A such that the topology on A induced from A is t -adic and t becomes aunit in A . A is called a ring of definition and t is called a pseudouniformizer . Let A be a Tatering. Then the pair ( A, A + ) is called an affinoid Tate ring , if A + ⊂ A is an open and integrallyclosed subring.For a Tate ring A , we denote by A ◦ ⊂ A the subset consisting of powerbounded elements of A and by A ◦◦ ⊂ A the subset consisting of topologically nilpotent elements of A . It is easy to verifythat A ◦◦ ⊂ A ◦ ⊂ A , A ◦ is a subring of A and A ◦◦ is an ideal of A ◦ . Let Spa( A, A + ) denote theset of continuous valuations | · | on an affinoid Tate ring ( A, A + ) satisfying an additional condition | A + | ≤ | · | ∈ Spa(
A, A + ). Then | · | factors through the domain A + / p , where p is the set of x ∈ A + for which | x | = 0; see Lemma 4.8. Then | · | defines a valuation ring VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 7 V ⊂ Frac( A + / p ). This valuation ring is microbial attached to | · | in view of [8, Proposition 7.3.7].For microbial valuation rings, we refer the reader to [35]. Let us recall the notion of perfectoidalgebras over a perfectoid field as defined in [50]. Definition 2.3 (Perfectoid K -algebra) . Fix a perfectoid field K and let A be a Banach K -algebra.Then we say that A is a perfectoid K-algebra , if the following conditions hold:(1) The set of powerbounded elements A ◦ ⊂ A is open and bounded.(2) The Frobenius endomorphism on A ◦ / ( p ) is surjective.We will recall the almost variant of perfectoid algebras; see [1]. Definition 2.4 (Almost perfectoid K -algebra) . Fix a perfectoid field K and let A be a Banach K -algebra with a basic setup ( A ◦ , I ). Then we say that A is I-almost perfectoid , if the followingconditions hold:(1) The set of powerbounded elements A ◦ ⊂ A is open and bounded.(2) The Frobenius endomorphism F rob : A ◦ / ( p ) → A ◦ / ( p ) is I -almost surjective. Example . Let A be a perfectoid K -algebra with a nonzero nonunit element t ∈ K ◦ admittinga compatible system of p -power roots { t pn } n> . Fix any nonzero divisor g ∈ A ◦ that admits acompatible system { g pn } n> . Let I := S n> ( tg ) pn . Then the pair ( A ◦ , I ) gives a basic setup,which is a prototypical example that is encountered in this article.The following notion is due to Artin [5]. Definition 2.6 (Absolute integral closure) . Let A be an integral domain. Then the absolute integralclosure of A denoted by A + , is defined to be the integral closure of A in a fixed algebraic closureof Frac( A ).The symbol for the absolute integral closure should not be confused with affinoid Tate ring( A, A + ). Let A be a ring with an element t ∈ A . Then we will denote by b A the t -adic completionof A . In most cases that we encounter, t will be either a nonzero divisor or t = p , a fixed primenumber. We say that a commutative ring A is normal , if the localization A p is an integrally closeddomain in its field of fractions for every prime ideal p ⊂ A . For ring maps A → C and B → C , wewrite A × C B for the fiber product. The completion of a module is always taken to be completeand separated. We also consider non-adic completion when studying Banach rings. We make useof Galois theory of commutative rings in making reductions in steps of proofs. Let A → B be aring extension and let G be a finite group acting on B as ring automorphisms. Then we say that B is a G-Galois extension of A , if A = B G and the natural ring map B ⊗ A B → Y G B ; b ⊗ b ′ ( γ ( b ) b ′ ) γ ∈ G is an isomorphism. Some fundamental results about Galois extensions are found in [1] or [19]. Adefinition of almost G-Galois extension is found in [1]. Definition 2.7.
Let A be ring with an element t ∈ A . Then we say that A is t-adically Zariskian ,if t is contained in every maximal ideal of A .3. Preliminary lemmas
Complete integral closure under completion.
We begin with definitions of closure op-erations of rings.
K.NAKAZATO AND K.SHIMOMOTO
Definition 3.1.
Let R ⊂ S be a ring extension.(1) An element s ∈ S is integral over R , if P ∞ n =0 R · s n is a finitely generated R -submodule of S . The set of all elements denoted as T of S that are integral over R forms a subring of S .If R = T , then R is called integrally closed in S .(2) An element s ∈ S is almost integral over R , if P ∞ n =0 R · s n is contained in a finitely generated R -submodule of S . The set of all elements denoted as T of S that are almost integral over R forms a subring of S , which is called the complete integral closure of R in S . If R = T ,then R is called completely integrally closed in S .From the definition, it is immediate to see that if R is a Noetherian domain and S is thefield of fractions of R , then R is integrally closed if and only if it is completely integrally closed.There are subtle points that we must be careful about on complete integral closure. The completeintegral closure T of R is not necessarily completely integrally closed in S and such an examplewas constructed by W. Heinzer [25]. Let R ⊂ S ⊂ T be ring extensions. Let b ∈ S be an element.Assume that b is almost integral over R when b is regarded as an element of T . Then it does notnecessarily mean that b is almost integral over R when b is regarded as an element of S ; see [23] forsuch an example. The following result is a key for the main results of [7]; see also [57, Tag 0BNR]for a proof and related results. Lemma 3.2 (Beauville-Laszlo) . Let A be a ring with a nonzero divisor t ∈ A and let b A be the t -adic completion. Then t is a nonzero divisor of b A and one has the commutative diagram: A −−−−→ b A y y A [ t ] −−−−→ b A [ t ] that is cartesian. In other words, we have A ∼ = A [ t ] × b A [ t ] b A . The following lemma is quite useful and often used in basic theory of perfectoid spaces. We takea copy from Bhatt’s lecture notes [8].
Lemma 3.3.
Let A be a ring with a nonzero divisor t ∈ A and let b A be the t -adic completion of A . Fix a prime number p > . Then the following assertions hold.(1) Suppose that A is integrally closed in A [ t ] . Then b A is integrally closed in b A [ t ] . If moreover A admits a compatible system of p -power roots { t pn } n> , then t − p ∞ A is integrally closedin A [ t ] .(2) Suppose that A is completely integrally closed in A [ t ] and A admits a compatible systemof p -power roots { t pn } n> . Then b A is completely integrally closed in b A [ t ] , and t − p ∞ A iscompletely integrally closed in A [ t ] .Proof. We refer the reader to [8, Lemma 5.1.1, Lemma 5.1.2 and Lemma 5.1.3]. Here we point outthat Lemma 3.2 plays a role in the proofs. (cid:3)
The following lemma is easy to prove, but plays an important role in our arguments.
Lemma 3.4.
Let A be a ring with a nonzero divisor t ∈ A such that A is completely integrallyclosed in A [ t ] . Fix a prime number p > . Suppose that A admits a compatible system of p -powerroots { t pn } n> . Then we have t − p ∞ A = A (in particular, t − p ∞ A is completely integrally closed in A [ t ] ). VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 9
Proof.
Since clearly A ⊂ t − p ∞ A , it suffices to show the reverse inclusion. Pick an element b ∈ t − p ∞ A . Then for every n >
0, there exists a n ∈ A such that t pn b = a n and therefore, b n = t − ( t pn ) p n − n a nn ∈ A [ t ]. Here notice that ( t pn ) p n − n a nn ∈ A . Thus, one obtains tb n ∈ A for every n > b ∈ A , because A is completely integrally closed in A [ t ]. (cid:3) Now let us discuss complete integral closedness of inverse limits.
Lemma 3.5.
Let A be a ring with an element t ∈ A , let Λ be a directed poset, and let { A λ } λ ∈ Λ an inverse system of A -algebras. Suppose that each A λ is t -torsion free and completely integrallyclosed in A λ [ t ] . Then lim ←− λ A λ is a t -torsion free A -algebra and completely integrally closed in (lim ←− λ A λ )[ t ] .Proof. Clearly, lim ←− λ A λ is a t -torsion free A -algebra. Pick an element b ∈ (lim ←− λ A λ )[ t ] which isalmost integral over lim ←− λ A λ . Then there exists some m > t m b n ∈ lim ←− λ A λ for every n >
0. Take d > a = ( a λ ) ∈ lim ←− λ A λ for which t d b = a . Then for every n >
0, it follows that t dn + m b n = t m a n , which implies t m a n ∈ t dn (lim ←− λ A λ ). Thus for each λ ∈ Λ, the element a λ t d ∈ A λ [ t ]satisfies t m ( a λ t d ) n ∈ A λ for every n . Since A λ is completely integrally closed in A λ [ t ], one finds that a λ ∈ t d A λ ( ∀ λ ∈ Λ) and thus a ∈ t d (lim ←− λ A λ ). Hence b = at d ∈ lim ←− λ A λ , as desired. (cid:3) In the situation of Lemma 2.1, complete integral closedness is preserved under ( ) ∗ . Lemma 3.6.
Let
A ֒ → B be a ring extension such that A is completely integrally closed in B .Suppose that A has an element t such that B is t -torsion free and A admits a compatible system of p -power roots { t pn } n> . Then t − p ∞ A is completely integrally closed in t − p ∞ B .Proof. Pick an element c ∈ t − p ∞ B which is almost integral over t − p ∞ A . We would like to showthat t pk c ∈ A for every k >
0. Since A is completely integrally closed in B , it suffices to checkthat each t pk c ∈ B is almost integral over A . Now by assumption, P ∞ n =0 t − p ∞ A · c n is containedin a finitely generated t − p ∞ A -submodule of t − p ∞ B . Hence t pk ( P ∞ n =0 t − p ∞ A · c n ) is contained ina finitely generated A -submodule of B for every k >
0. Meanwhile, it follows that ∞ X n =0 A · ( t pk c ) n ⊂ t pk (cid:18) ∞ X n =0 A · c n (cid:19) ⊂ t pk (cid:18) ∞ X n =0 t − p ∞ A · c n (cid:19) . Therefore, t pk c ∈ B is almost integral over A , as desired. (cid:3) Lemma 3.7.
The following assertions hold.(1) Let R be a Noetherian integrally closed domain with its absolute integral closure R + andassume that A is a ring such that R ⊂ A ⊂ R + . Then A is integrally closed in Frac( A ) ifand only if A is completely integrally closed in Frac( A ) .(2) Let R ⊂ S ⊂ T be ring extensions. Assume that R is completely integrally closed in T .Then R is also completely integrally closed in S .Proof. (1): The proof is found in the proof of [53, Theorem 5.9], whose statement is given only forNoetherian normal rings of characteristic p >
0. However, the argument there remains valid forNoetherian normal rings of arbitrary characteristic.(2): For s ∈ S , assume that P ∞ n =0 R · s n is contained in a finitely generated R -submodule of S .Then this property remains true when regarded as an R -submodule of T . So we have s ∈ R by ourassumption. (cid:3) Lemma 3.8.
Let A be a normal domain with field of fractions Frac( A ) and assume that Frac( A ) ֒ → B is an integral extension such that B is reduced. Denote by C := A + B the integral closure of A in B . Then C p is a normal domain for any prime ideal p of C .Proof. Notice that B can be written as the filtered colimit of finite integral subextensions Frac( A ) → B ′ → B . Without loss of generality, we may assume and do that Frac( A ) → B is a finite integralextension. Since Frac( A ) is a field, B is a reduced Artinian ring, so that we can write B = Π mi =1 L i with L i being a field. Since A → C is torsion free and integral, we see that Frac( A ) ⊗ A C is thetotal ring of fractions of C , which is just B . In other words, C has finitely many minimal primeideals, because so does B . Then by [57, Tag 030C], C is a finite product of normal domains, whichshows that C p is a normal domain for any prime ideal p ⊂ C . (cid:3) Galois theory of rings is closely related to a study of integrality. For example, the followingstatement holds.
Lemma 3.9.
Let A be a Tate ring and let A ֒ → B be a Galois extension with Galois group G .Equip B with the canonical structure as a Tate ring as in [44, Lemma 2.17] . Then the action of G preserves B ◦ . Moreover, if further A is preuniform (see [44, Definition 2.12 and Definition 2.14] ),then ( B ◦ ) G = A ◦ .Proof. Let A be a ring of definition of A and let t ∈ A be a pseudouniformizer of A . As in theproof of [44, Lemma 2.17], we can take a ring of definition B of B that is finitely generated asan A -module and satisfies B = B [ t ]. Pick b ∈ B ◦ and σ ∈ G arbitrarily. Then there is some l > t l b n ∈ B and therefore, t l σ ( b ) n ∈ σ ( B ) for every n >
0. Meanwhile, since σ ( B )is also finitely generated as an A -module, we have t l ′ σ ( B ) ⊂ B for some l ′ >
0. Hence σ ( b ) isalso almost integral over B . Thus, the action of G preserves B ◦ . If further A is preuniform, thenwe have ( B ◦ ) G = B G ∩ B ◦ = A ∩ B ◦ = A ◦ by [44, Corollary 4.8(4)], as wanted. (cid:3) A variant of perfectoid Abhyankar’s lemma for almost Witt-perfect rings
Let p >
Witt-perfect rings due to Davis and Kedlaya; see [16] and [17].
Definition 4.1 (Witt-perfect ring) . For a prime number p >
0, we say that a p -torsion free ring A is p-Witt-perfect (simply Witt-perfect ), if the Witt-Frobenius map F : W p n ( A ) → W p n − ( A ) issurjective for all n > Definition 4.2 (Almost Witt-perfect ring) . Let A be a p -torsion free ring with an element π ∈ A admitting a compatible system of p -power roots π pn ∈ A . Then we say that A is ( π ) p ∞ -almostWitt-perfect , if the following conditions are satisfied.(1) The Frobenius endomorphism on A/ ( p ) is ( π ) p ∞ -almost surjective.(2) For every a ∈ A and every n >
0, there is an element b ∈ A such that b p ≡ pπ pn a (mod p ).In applications, we often consider the case that π ∈ A is a nonzero divisor and A is (completely)integrally closed in A [ p ]. If one takes π = 1, then it is shown that ( π ) p ∞ -almost Witt perfectnesscoincides with the Witt-perfectness; see [44] for details. Let us recall the following fact; see [44,Proposition 3.20]. VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 11
Proposition 4.3.
Let V be a p -adically separated p -torsion free valuation ring and let A be a p -torsion free V [ T p ∞ ] -algebra. Set π pn := T pn · ∈ A for every n ≥ and denote by b V and b A the p -adic completions of V and A , respectively. Then the following conditions are equivalent. ( a ) V is a Witt-perfect valuation ring of rank and A is ( π ) p ∞ -almost Witt-perfect and inte-grally closed (resp. completely integrally closed) in A [ p ] . ( b ) There exist a perfectoid field K and a ( π ) p ∞ -almost perfectoid K h T p ∞ i -algebra A with thefollowing properties: • K is a Banach ring associated to ( b V , ( p )) , the norm on K is multiplicative and K ◦ = b V ; • A is a Banach ring associated to ( b A, ( p )) and b A is open and integrally closed in A (resp. A ◦ = b A ); • the bounded homomorphism of Banach rings K h T p ∞ i → A is induced by the ring map V [ T p ∞ ] → A . Remark 4.4. (1) In Proposition 4.3, one is allowed to map T to pg ∈ A , in which case A isa ( pg ) p ∞ -almost Witt-perfect ring for some g ∈ A . We will consider Witt-perfect rings ofthis type.(2) The advantage of working with (almost) Witt-perfect rings is in the fact that one need notimpose p -adic completeness condition on a ring. Let A := W ( k )[[ x , . . . , x d ]] be the powerseries algebra over the ring of Witt vectors of a perfect field k of characteristic p >
0. Then A ∞ := [ n> W ( k )[ p pn ][[ x pn , . . . , x pn d ]]is a Witt-perfect algebra that is an integrally closed domain and integral, faithfully flatover A . The p -adic completion d A ∞ of A ∞ is integral perfectoid. While A → d A ∞ remainsflat, it is not integral. The ring A ∞ will be used essentially in the construction of almostCohen-Macaulay algebras later. A similar construction for complete ramified regular localrings appears in [54, Proposition 4.9].Another important example of a Witt-perfect ring is given by an arbitrary absolutelyintegrally closed domain A , where A is a faithfully flat Z p -algebra. The p -adic completion b A is an integral perfectoid algebra over \Z p [ p p ∞ ]. Indeed, as A is absolutely integrally closedin its field of fractions, it contains Z + p . Hence b A is a c Z + p -algebra. Lemma 4.5.
Assume that A is a ring with a regular sequence ( a, b ) . Let b A a denote the a -adiccompletion of A . Then a and b are nonzero divisors of b A a .Proof. That a is a nonzero divisor was already mentioned in Lemma 3.2. Let t ∈ b A a be suchthat bt = 0. Then one obviously has bt ∈ a n b A a for all n >
0. Since b is a nonzero divisor on A/ ( a n ) ∼ = b A a / ( a n ), it follows that t ∈ T n> a n b A a = 0 and thus, b is a nonzero divisor on b A a . (cid:3) Example . Let us consider the subring: R := Z (cid:2) xp , xp , . . . (cid:3) ⊂ Q [ x ] . Then it is clear that R is a domain. However, after taking the p -adic completion b R , since x ∈ p n R , x becomes zero in b R . Therefore, p is a nonzero divisor of b R , while x is not so. Variants of Riemann’s extension theorems.
Let us explain the
Riemann’s extensiontheorem in the language of commutative algebra. This is a key result to the proof of the DirectSummand Conjecture and its derived variant; see [2] and [9]. For simplicity, let A be a ring withnonzero divisors f, g ∈ A . Then we can consider the subring A [ f n g ] in A [ g ]. In other words, wedefine A [ f n g ] := (cid:0) A [ T ] / ( gT − f n ) (cid:1)(cid:14) a , where a := S m> (0 : g m ) as an ideal of A [ T ] / ( gT − f n ). Problem 1 (Algebraic formulation of Riemann’s extension problem) . Study the ring-theoreticstructure of the intersection \ n> A [ f n g ] , where A [ f n +1 g ] → A [ f n g ] is the natural inclusion defined by f n +1 g f · f n g . Notice that the intersectionis taken inside A [ g ] . In his remarkable paper [51], Scholze studied the perfectoid version of the above problem, withan application to the construction of Galois representations using torsion classes in the cohomologyof certain symmetric spaces. Before going further, we need the notion of semivaluations . Definition 4.7.
Let A be a ring and let | · | : A → Γ ∪ { } be a map for a totally ordered abeliangroup Γ with group unit 1 and we let 0 < γ for arbitrary γ ∈ Γ. Then | · | is called a semivaluation ,if | | = 0, | | = 1, | xy | = | x || y | and | x + y | ≤ max {| x | , | y |} for x, y ∈ A .The name semivaluation refers to the fact that A need not be an integral domain. Those semi-valuations that satisfy a certain topological condition are called continuous , which are extensivelystudied in Huber’s papers [33] and [34]. Lemma 4.8.
Let | · | : A → Γ ∪ { } be a semivaluation. Then p := { x ∈ A | | x | = 0 } is a primeideal of A and if we set k ( p ) := Frac( A/ p ) , then | · | induces a valuation | · | p : k ( p ) → Γ ∪ { } .Proof. This is an easy exercise, using the properties stated in Definition 4.7. (cid:3)
The prime ideal p in Lemma 4.8 is called the support of the semivaluation | · | . For a given | · | : A → Γ ∪ { } , we set V |·| := { x ∈ k ( p ) | | x | p ≤ } . Then this is a valuation ring with its field offractions k ( p ). Definition 4.9.
Let D ⊂ C be a ring extension and let us setVal( C, D ) := n | · | (cid:12)(cid:12)(cid:12) | · | is a semivaluation on C such that | D | ≤ V |·| has dimension ≤ o. ∼ , where ∼ is generated by natural equivalence classes of semivaluations.Let us prove the following algebraic lemma. Lemma 4.10.
Let ( C, D ) be a pair of rings such that C is the localization of D with respect tosome multiplicative set consisting of nonzero divisors. Suppose that D is an integral extension ofa Noetherian ring R . Fix a (possible empty) subset S ⊂ D that consists of only nonzero divisors.Then one has D + C = n x ∈ C (cid:12)(cid:12)(cid:12) | x | ≤ for any | · | ∈ Val(
C, D ) such that | g | 6 = 0 for every g ∈ S o , where D + C is the integral closure of D in C . VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 13
Proof.
Since the containment ⊂ is clear by the definition of Val( C, D ), let us prove the reversecontainment ⊃ . Let y ∈ C be such that | y | ≤
1, where | · | ∈
Val(
C, D ) satisfies | g | 6 = 0 for every g ∈ S . Let D [ y ] be the subring of the localization C [ y ] which is generated by y − = y over D . Consider the ring extension D [ y ] ⊂ C [ y ]. First suppose that y − is a unit in D [ y ]. Then we canwrite y = a y n − + a y n − + · · · + a n − for a i ∈ D . Then we have y n − a n − y n − − · · · − a = 0. Hence y ∈ C is integral over D and y ∈ D + C .To derive a contradiction, suppose that y − ∈ D [ y ] is not a unit. We may assume that y isnot nilpotent. Choose a prime ideal m ⊂ D [ y ] such that y − ∈ m . Let p ⊂ D [ y ] be a minimalprime ideal satisfying p ⊂ m . On the other hand, R [ y ] ⊂ D [ y ] is an integral extension and R [ y ] isNoetherian by Hilbert’s Basis Theorem.Then, one can find a valuation ring D [ y ] / p ⊂ V ⊂ Frac( D [ y ] / p ) such that the center (the maxi-mal ideal) of V contains y − and the Krull dimension of V is 1: More concretely, one can construct V in the following way. Let n := m ∩ R [ y ] and q := p ∩ R [ y ]. Then we have a Noetherian subdomain R [ y ] / q ⊂ Frac( R [ y ] / q ). By [58, Theorem 6.3.2 and Theorem 6.3.3], there is a Noetherian valuationring V n such that R [ y ] / q ⊂ V n ⊂ Frac( R [ y ] / q ) and the center of V n contains n ⊂ R [ y ] / q . We havethe commutative diagram: Frac( R [ y ] / q ) −−−−→ Frac( D [ y ] / p ) x x V n −−−−→ V where V is defined as the localization of the integral closure of V n in Frac( D [ y ] / p ) (this integralclosure is a so-called Pr¨ufer domain ) at the maximal ideal containing m . So V is a valuation ringof Krull dimension 1 and we have the composite map D → D [ y ] → V . Let | · | V denote thecorresponding valuation. Moreover, S ⊂ D consists of nonzero divisors and C [ y ] is the localizationof D , so the image of elements in S remains nonzero divisors in C [ y ] and thus in the subring D [ y ].As p is a minimal prime ideal of D [ y ], g / ∈ p for every g ∈ S . So we find that | g | V = 0 and inparticular, this implies that D → D [ y ] → V extends to the map C → C [ y ] → Frac( V ) and thesemivaluation on ( C, D ) induced by | · | V gives a point | · | C ∈ Val(
C, D ).By our assumption, we have | y | C ≤
1. Since y − ∈ V is in the center, we know | y − | C < | y | C | y − | C = | yy − | C = 1 and thus, y − ∈ D [ y ] mustbe a unit, as desired. (cid:3) The above lemma has the following implication: Keep in mind that A + stands for an openintegrally closed subring in a Tate ring A . Corollary 4.11.
Let ( A, A + ) be an affinoid Tate ring with a fixed pseudouniformizer t ∈ A + suchthat A + is t -adically Zariskian and A + is integral over a Noetherian ring. For a nonzero divisor g ∈ A + , let us set ( C, D ) := ( A [ g ] , A + ) . Then we have (4.1) D + C = n x ∈ C (cid:12)(cid:12)(cid:12) | x | ≤ for any | · | ∈ Val(
C, D ) such that | t | < o . Notice that D [ y ] is not the localization of D with respect to the multiplicative system { y n } n ≥ . Finally, let
Val(
C, D ) | t | < be the set of all elements | · | ∈ Val(
C, D ) for which < | t | < . Thenthe natural map ( A, A + ) → ( C, D ) induces an injection Val(
C, D ) | t | < ֒ → Spa(
A, A + ) .Proof. Keep the notation as in the proof of Lemma 4.10. The point is that one can choose thevaluation domain V so as to satisfy the required property. So assume that y ∈ A [ g ] satisfies | y | ≤ | · | ∈ Val( A [ g ] , A + ) and y − ∈ A + [ y ] is not a unit. Then we can find a maximal ideal m ⊂ A + [ y ] such that y − ∈ m , which gives the surjection A + ։ A + [ y ] / m and let n ⊂ A + beits kernel. Then n is a maximal ideal of A + . The element t ∈ A + is in the Jacobson radicalby assumption, so we have t ∈ n . There is a chain of prime ideals p ⊂ m ⊂ A + [ y ] such that p is minimal and t, y − ∈ m . Then, we have the associated valuation ring ( V, | · | V ) and the map A + [ y ] / p ֒ → V . It follows from the above construction that | t | V <
1, establishing (4 . t mapsinto the maximal ideal of the rank 1 valuation ring V , it follows from [8, Proposition 7.3.7] that | · | V pulled back to A + gives a point of Spa( A, A + ). Finally, the injectivity of the claimed map isclear from the construction. (cid:3) Corollary 4.11 can be formulated also in terms of adic geometry, as follows.
Corollary 4.12.
Let ( A, A + ) be an affinoid Tate ring and let ( A , ( t )) be a pair of definition of A .Let s ∈ A be an element such that t ∈ sA . Let X = Spa( A, A + ) and let U be the subspace of X : U := n x ∈ X (cid:12)(cid:12)(cid:12) | s | e x < for the maximal generization e x of x o . Suppose that A is s -adically Zariskian and integral over a Noetherian ring. Then we have A + = A ◦ = ( A ) + A = n a ∈ A (cid:12)(cid:12)(cid:12) | a | x ≤ for any x ∈ [ U ] o , where [ U ] denotes the maximal separated quotient of U .Proof. Since we have the containments( A ) + A ⊂ A + ⊂ A ◦ ⊂ n a ∈ A (cid:12)(cid:12)(cid:12) | a | x ≤ x ∈ [ U ] o (the third inclusion holds because | · | x is of rank 1), it suffices to show that(4.2) ( A ) + A = n a ∈ A (cid:12)(cid:12)(cid:12) | a | x ≤ x ∈ [ U ] o . By assumption, there exists g ∈ A such that t = sg . Let B be the Tate ring associated to ( A , ( s ))and B + := ( A ) + B . Then we have A = B [ g ], ( A ) + A = ( B + ) + B [ g ] and(4.3) ( B + ) + B [ g ] = n b ∈ B [ g ] (cid:12)(cid:12)(cid:12) | b | ≤ | · | ∈ Val( B [ g ] , B + ) | s | < o by Corollary 4.11. Let us deduce (4.2) from (4 .
3) by constructing a canonical bijectionVal( B [ g ] , B + ) | s | < ∼ = −→ [ U ] . Any point | · | ∈
Val( B [ g ] , B + ) | s | < satisfies that | a | ≤ a ∈ A and | t | = | sg | <
1. Thus,since | · | is of rank 1, | · | gives a continuous semivaluation on A such that | A ◦ | ≤
1. Hence we havea canonical injection Val( B [ g ] , B + ) | s | < ֒ → [ U ] . Moreover, it is surjective because B + ⊂ A + and s ∈ A is invertible. Thus the assertion follows. (cid:3) Indeed, the following immediate corollary is already documented in a treatise on rigid geometry.
VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 15
Corollary 4.13 (cf. [20, II , Theorem 8.1.11 and 8.2.19]) . Let A be a complete and separated Tatering. Suppose that A has a ring of definition A that is Noetherian. Set X = Spa (cid:0) A, ( A ) + A (cid:1) . Thenwe have ( A ) + A = (cid:8) a ∈ A (cid:12)(cid:12) | a | x ≤ x ∈ [ X ] (cid:9) . Now we can establish a weak form of Riemann’s extension theorem, which is fitting into theframework of Zariskian geometry; see [59] for more details.
Theorem 4.14 (Riemann’s extension theorem I) . Let A be a ring with a nonzero divisor ̟ that is ̟ -adically Zariskian and integral over a Noetherian ring. Let g ∈ A be a nonzero divisor. Let A j the Tate ring associated to (cid:0) A [ ̟ j g ] , ( ̟ ) (cid:1) for every integer j > . Then we have an isomorphism ofrings A + A [ ̟g ] ∼ = −→ lim ←− j A j ◦ , where the transition map A j +1 ◦ → A j ◦ is the natural one.Proof. By assumption, we have a canonical ring isomorphism ϕ j : A [ ̟g ] ∼ = −→ A j for each j >
0. Byrestricting ϕ j to A + A [ ̟g ] , we obtain the ring map ϕ + j : A + A [ ̟g ] → A j ◦ . Then { ϕ j } j> and { ϕ + j } j> induce the commutative diagram of ring maps(4.4) A + A [ ̟g ] ϕ + −−−−→ lim ←− j A j ◦ y y A [ ̟g ] ∼ = −−−−→ ϕ lim ←− j A j where ϕ is an isomorphism and the vertical maps are injective. Thus it suffices to prove that (4.4)is cartesian. Pick c ∈ A [ ̟g ] such that ϕ j ( c ) ∈ A j ◦ for every j >
0. Let us show that c lies in A + A [ ̟g ] by applying Corollary 4.12. For this, we consider the ( ̟g )-adic topology: let A ( ̟g ) be the Tatering associated to (cid:0) A, ( ̟g ) (cid:1) (notice that each A j is also the Tate ring associated to (cid:0) A [ ̟ j g ] , ( ̟g ) (cid:1) ).Let X ( ̟g ) = Spa( A ( ̟g ) , A + A ( ̟g ) ), X j = Spa( A j , A j ◦ ) for each j >
0, and U be the subspace U = n x ∈ X ( ̟g ) (cid:12)(cid:12)(cid:12) | ̟ | e x < e x of x o of X ( ̟g ) . Then the underlying ring of A ( ̟g ) is equal to A [ ̟g ], and we have A + A ( ̟g ) = n a ∈ A ( ̟g ) (cid:12)(cid:12)(cid:12) | a | x ≤ x ∈ [ U ] o by Corollary 4.12. On the other hand, since we also have A j ◦ = n a ∈ A j (cid:12)(cid:12)(cid:12) | a | x j ≤ x j ∈ [ X j ] o by Proposition 6.1, | ϕ j ( c ) | x j ≤ j > x j ∈ [ X j ]. Now since ϕ j gives a continuousmap (cid:0) A ( ̟g ) , ( A ) + A ( ̟g ) (cid:1) → ( A j , A j ◦ ), (4.4) induces the continuous map lim −→ j [ X j ] → [ X ( ̟g ) ], whichfactors through [ U ] because ̟ ∈ A j is topologically nilpotent. Thus we are reduced to showingthat the resulting map f : lim −→ j [ X j ] → [ U ] is surjective. Pick x ∈ [ U ] and let | · | x : A ( ̟g ) → R ≥ be a corresponding semivaluation. Let us find some j > | · | x,j : A j → A ( ̟g ) | · | x −−→ R ≥ gives a point x j ∈ [ X j ] for which f ([ x j ]) = x . Since | ̟ | x < | · | x is of rank 1, there existssome j > | ̟ j g | x <
1. Then we have | A [ ̟ j g ] | x ≤ | A | x ≤ | · | x is ofrank 1. Thus, since any a ∈ A j ◦ is almost integral over A [ ̟ j g ] and | · | x is of rank 1, we have | A j ◦ | x,j ≤
1. Hence | · | x,j gives the desired point x j ∈ [ X j ]. (cid:3) We shall investigate the Riemann’s extension problem in the context of Witt-perfect rings bytransporting the situation to the case of perfectoid algebras, in which case Riemann’s extensiontheorem has been studied by Andr´e, Bhatt and Scholze and known to experts. Although thestatement of Theorem 4.15 below has a partial overlap with Theorem 4.14, we decided to giveanother proof to Theorem 4.15 by making the reduction to the already-known Riemann’s exten-sion theorem for perfectoid algebras upon p -adic completion. Let us start setting up some notation. Notation : Fix a prime number p > p -torsion free ring A such that p is not a unitin A and admits a compatible system of p -power roots g pn ∈ A for a nonzero divisor g ∈ A for n >
0. Moreover, assume that A is faithfully flat over a Witt-perfect valuation domain V of rank1 such that p pn ∈ V for n > A is a ( pg ) p ∞ -almost Witt-perfect ring and A iscompletely integrally closed in A [ p ]. Let us put(4.5) A := b A [ 1 p ] and K := b V [ 1 p ] , where the completion is p -adic.If ( p, g ) is a regular sequence, then it follows from Lemma 4.5 that g ∈ A is a nonzero divisorand A equipped with some norm (associated to ( b A, ( p ))) is a ( pg ) p ∞ -almost perfectoid algebra overthe perfectoid field K by Proposition 4.3. The natural homomorphism A ♮ ֒ → A is a ( pg ) p ∞ -almostisomorphism, where A ♮ denotes the untilt of the tilt of A and A ♮ is a perfectoid K -algebra in viewof [1, Proposition 3.5.4]. For each j >
0, we define an A -algebra:(4.6) A j := A (cid:8) p j g (cid:9) := Ah T i / ( gT − p j ) − where Ah T i is the completion of A [ T ] with respect to the Gauss norm (cf. [38, Definition 1.6]) and( gT − p j ) − is the closure of the ideal ( gT − p j ) in Ah T i . We equip A j with the quotient norm. Then A j is a Banach A -algebra and it is viewed as a ring of analytic functions on the rational subset (cid:8) x ∈ X (cid:12)(cid:12) | p j | ≤ | g ( x ) | (cid:9) of X := Spa( A , A ◦ ) (cf. [34, Proposition 1.3 and 1.6]). Moreover, because A ♮ [ g ] ∼ = A [ g ] and p ∈ A ♮ is a unit, we have ( A ♮ ) j ∼ = A j as topological rings for all j >
0. We havea ( pg ) p ∞ -almost isomorphism:(4.7) \ A ♮ ◦ (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) ≈ A j ◦ VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 17 in view of Scholze’s result [50, Lemma 6.4]. Since b A ∼ = A ◦ and A ♮ ◦ ≈ b A is a ( pg ) p ∞ -almostisomorphism, (4 .
7) factors as \ A ♮ ◦ (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) → \ A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) → A j ◦ , which yields a ( pg ) p ∞ -almost isomorphism:(4.8) \ A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) ≈ A j ◦ . Then A j ◦ is bounded and it is an integral perfectoid K ◦ -algebra. The set of A -algebras {A j } j> forms an inverse system, where A j +1 → A j is the natural inclusion defined by(4.9) T pT. Then A j +1 → A j is a continuous map between Banach K -algebras, so that it induces A j +1 ◦ → A j ◦ .Let ψ : A [ pg ] → A j be the natural map. Set(4.10) A j := n a ∈ A [ 1 pg ] (cid:12)(cid:12)(cid:12) ψ ( a ) ∈ A j ◦ o ∼ = A [ 1 pg ] × A j A j ◦ , where the second isomorphism follows from the injectivity of A j ◦ → A j . Set A j to be a Tate ringwhose underlying ring is A [ pg ](= A j [ p ]) such that { p n A j } n ≥ forms a fundamental system of openneighborhoods of 0 ∈ A j . Then, since A j ◦ is a ring of definition of A j , A j ◦ = A j and thus A j ◦ iscompletely integrally closed in A [ pg ]. We have the induced maps A j +1 ◦ → A j ◦ in view of (4 . A −−−−→ A j +1 ◦ (cid:13)(cid:13)(cid:13) y A −−−−→ A j ◦ After the preparations we have made above, let us establish
Witt-perfect Riemann’s ExtensionTheorem . Theorem 4.15 (Riemann’s extension theorem II) . Let A be a p -torsion free ring that is flat over aWitt-perfect valuation domain V of rank admitting a compatible system of p -power roots p pn ∈ V ,together with a nonzero divisor g ∈ A admitting a compatible system of p -power roots g pn ∈ A .Suppose that(1) A is p -adically Zariskian and A is completely integrally closed in A [ p ] ;(2) A is a ( pg ) p ∞ -almost Witt-perfect ring that is integral over a Noetherian ring;(3) ( p, g ) is a regular sequence on A (or more generally, it suffices to assume that p, g arenonzero divisors on the p -adic completion b A ).Let us put e A := lim ←− j> A j ◦ . Then the following statements hold. (a) There is a ( pg ) p ∞ -almost isomorphism for each j > : A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3) ≈ A j ◦ . (b) The natural map A j ◦ → A j ◦ induces an isomorphism: d A j ◦ ∼ = −→ A j ◦ . In particular, A j ◦ is Witt-perfect.(c) There are ring isomorphisms: A + A [ pg ] ∼ = A ∗ A [ pg ] ∼ = e A , where A + A [ pg ] (resp. A ∗ A [ pg ] ) isthe integral closure (resp. complete integral closure) of A in A [ pg ] . Moreover, if A iscompletely integrally closed in A [ pg ] , then A ∼ = g − p ∞ A ∼ = e A .Proof. Before starting the proof, let us remark that b A ∼ = A ◦ , because A is completely integrallyclosed in A [ p ]. The elements p, g ∈ A ◦ are nonzero divisors in view of Lemma 4.5 and A ◦ is anintegral ( pg ) p ∞ -almost perfectoid b V -algebra.First, we prove the assertion ( a ). By taking the functor of almost elements ( pg ) − p ∞ ( ), it sufficesto prove that ( pg ) − p ∞ (cid:16) A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3)(cid:17) → ( pg ) − p ∞ A j ◦ ∼ = A j ◦ is an honest isomorphism. Here notice that ( pg ) − p ∞ A j ◦ ∼ = A j ◦ is the consequence from Lemma 3.4and complete integral closedness of A j ◦ in A j ◦ [ pg ]. By taking the p -adic completion, we get(4.11) \ ( pg ) − p ∞ (cid:16) A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3)(cid:17) ∼ = ( pg ) − p ∞ (cid:16) \ A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3)(cid:17) ∼ = A j ◦ , where the first isomorphism follows from the fact that the functor of almost elements commuteswith the completion functor by Lemma 2.1, and the second one is due to (4 . A [ pg ] = ( pg ) − p ∞ (cid:16) A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3)(cid:17) [ p ], it follows from (4 .
10) and (4 .
11) combined with Lemma 3.2 thatthere is an isomorphism:(4.12) A j ◦ ∼ = ( pg ) − p ∞ (cid:16) A (cid:2)(cid:0) p j g (cid:1) p ∞ (cid:3)(cid:17) , as desired.The proof of the assertion ( b ) is obtained by the combination of (4 .
11) and (4 . c ). First, note that e A ∼ = lim ←− j> A j ◦ ∼ = lim ←− j> n x ∈ A [ 1 pg ] (cid:12)(cid:12)(cid:12) ψ ( x ) ∈ A j ◦ o ∼ = lim ←− j> (cid:16) A [ 1 pg ] × A j A j ◦ (cid:17) (4.13) ∼ = A [ 1 pg ] × lim ←− j> A j (cid:0) lim ←− j> A j ◦ (cid:1) ∼ = n x ∈ A [ 1 pg ] (cid:12)(cid:12)(cid:12) e ψ ( x ) ∈ lim ←− j> A j ◦ o , where e ψ : A [ pg ] → A [ g ] is the natural map. One claims the following equality:(4.14) A + A [ pg ] = n x ∈ A [ 1 pg ] (cid:12)(cid:12)(cid:12) e ψ ( x ) ∈ g − p ∞ A ◦ o ∼ = A [ 1 pg ] × A [ g ] g − p ∞ A ◦ , VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 19 which we prove now. By Riemann’s extension theorem for (almost) perfectoid K -algebras [1,Th´eor`eme 4.2.2], we know an isomorphism:(4.15) g − p ∞ A ◦ ∼ = lim ←− j> A j ◦ . As A j ◦ is completely integrally closed in A j ◦ [ pg ], it follows that the right-hand side of (4 . pg by Lemma 3.5. This implies that g − p ∞ A ◦ iscompletely integrally closed after inverting pg . Thus, A + A [ pg ] is contained in the right-hand side of(4 . A → g − p ∞ A is almost integral andwe have A ⊂ g − p ∞ A ⊂ A + A [ pg ] by Proposition 6.1. So Corollary 4.11 gives us(4.16) A + A [ pg ] = n x ∈ C (cid:12)(cid:12)(cid:12) | x | ≤ ∀ | · | ∈ Val(
C, D ) | p | < o , by setting ( C, D ) := ( g − p ∞ A [ pg ] , g − p ∞ A ). Note that g − p ∞ A [ pg ] = A [ pg ]. Equip ( g − p ∞ A )[ p ] withthe canonical structure as a Tate ring by declaring that g − p ∞ A is a ring of definition and thetopology is p -adic. A result of Huber [33, Proposition 3.9] asserts that Val(
C, D ) | p | < ֒ → Spa (cid:16) ( g − p ∞ A )[ 1 p ] , g − p ∞ A (cid:17) ∼ = Spa (cid:16) ( g − p ∞ b A )[ 1 p ] , g − p ∞ b A (cid:17) ∼ = Spa (cid:16) ( g − p ∞ A ◦ )[ 1 p ] , g − p ∞ A ◦ (cid:17) , which shows that any | · | ∈ Val(
C, D ) | p | < extends to an element | · | ∈ Spa (cid:16) ( g − p ∞ A ◦ )[ p ] , g − p ∞ A ◦ (cid:17) for which we know | x | ≤ x ∈ g − p ∞ A ◦ . This fact combined with (4 .
16) yields the following: A + A [ pg ] ⊂ n x ∈ A [ 1 pg ] (cid:12)(cid:12)(cid:12) e ψ ( x ) ∈ g − p ∞ A ◦ o ⊂ n x ∈ C (cid:12)(cid:12)(cid:12) | x | ≤ ∀ | · | ∈ Val(
C, D ) | p | < o = A + A [ pg ] , so that (4 .
14) has been proved. Therefore, (4 . .
14) and Proposition 6.1 can be put togetherto derive the desired isomorphisms: A + A [ pg ] ∼ = A ∗ A [ pg ] ∼ = e A. The last assertion is now clear. (cid:3)
The following proposition is crucial in the proof of Witt-perfect Abhyankar’s lemma. Proposition 4.16.
Let the notation and hypotheses be as in Theorem 4.15. Then \ lim ←− j> A j ◦ ∼ = lim ←− j> d A j ◦ , where c ( ) is p -adic completion. Notice that ( g − p ∞ A )[ p ] may differ from g − p ∞ ( A [ p ]). But the former is contained in the latter and Lemma 3.6applies to claim that g − p ∞ A is an integrally closed subring of ( g − p ∞ A )[ p ]. Notice that in general, inverse limits and taking completion do not commute. This proposition seems to be inthe heart of the reduction of Riemann’s extension theorem for perfectoid algebras to its non-complete (Witt-perfect)version.
Proof.
By Riemann’s extension theorem for (almost) perfectoid K -algebras [1, Th´eor`eme 4.2.2], wehave(4.17) g − p ∞ A ◦ ∼ = lim ←− j> A j ◦ . Since the right-hand side of (4 .
17) is equal to the intersection of p -adically complete modules, itfollows that(4.18) lim ←− j> d A j ◦ ∼ = lim ←− j> A j ◦ is also p -adically complete and separated. On the other hand, we have a natural map:(4.19) lim ←− j> A j ◦ → lim ←− j> d A j ◦ . Notice that p n d A j ◦ ∩ A j ◦ = p n A j ◦ for n >
0. Since p is a nonzero divisor on both A j ◦ and d A j ◦ andintersection commutes with inverse limit, it follows that (cid:16) p n lim ←− j> d A j ◦ (cid:17) ∩ lim ←− j> A j ◦ = p n lim ←− j> A j ◦ . This says that the topology on lim ←− j> A j ◦ induced from the inverse image of the filtration { p n lim ←− j> d A j ◦ } n> via (4 .
19) coincides with the p -adic topology. So \ lim ←− j> A j ◦ is the topologi-cal closure of the image of (4 . \ lim ←− j> A j ◦ → lim ←− j> d A j ◦ isinjective. It remains to prove that this is surjective. Since A ֒ → A j ◦ and A j ◦ is completely integrallyclosed in A j ◦ [ pg ], Lemma 3.4 shows that g − p ∞ A ֒ → lim ←− j> A j ◦ , which induces a composite map(4.20) lim ←− j> d A j ◦ ∼ = g − p ∞ A ◦ ∼ = g − p ∞ b A ∼ = \ g − p ∞ A → \ lim ←− j> A j ◦ , where the first isomorphism follows from (4 .
17) and (4 . \ lim ←− j> A j ◦ → lim ←− j> d A j ◦ gives a splitting of the map (4 .
20) up to an isomorphism, and \ lim ←− j> A j ◦ → lim ←− j> d A j ◦ is surjective, as desired. (cid:3) Problem 2.
Suppose that ( A, A + ) is an affinoid Tate ring such that A + is almost Witt-perfect andcompletely integrally closed in A . Then is the pair ( A, A + ) sheafy, or is it stably uniform? Some relevant results are found in the papers [14] and [43].
Remark 4.17.
Witt-perfect rings are almost never Noetherian and thus, it is natural to askwhether such algebras could be integral over a Noetherian ring. One way for constructing suchan algebra over a Noetherian normal domain R is to take the maximal ´etale extension of R . Thedetails are found in [55] and [56] and we will apply this method effectively to construct almostCohen-Macaulay algebras. VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 21
Witt-perfect Abhyankar’s lemma.
Now we are prepared to prove the main theorem, whichis a Witt-perfect version of Andr´e’s Perfectoid Abhyankar’s Lemma. First, we need a lemma.
Lemma 4.18.
Keep the notation and hypotheses as in Theorem 4.19 below and let B [ p j g ] ⊂ B ′ bethe B -subalgebra that is generated by p j g . Then the following statements hold:(1) There is an isomorphism: A ∼ = −→ g − p ∞ A and in particular, B is equal to the integral closureof A in B ′ .(2) Let B j be the module-finite A j -algebra B ′ , which is equipped with the canonical structure asa Tate ring as in [44, Lemma 2.17] . Then (cid:0) B [ p j g ] , ( p ) (cid:1) is a pair of definition of B j .(3) Let B and B{ p j g } be the separated completions of Tate rings associated to (cid:0) B, ( p ) (cid:1) and (cid:0) B [ p j g ] , ( p ) (cid:1) , respectively and let B j be the module-finite A j -algebra B ′ ⊗ A [ pg ] A j equippedwith the canonical topology as in [44, Lemma 2.16] . Then we have a canonical isomorphismof topological rings: B j ∼ = −→ B{ p j g } . In particular, B j is complete and separated.(4) B is p -adically Zariskian and B is completely integrally closed in B [ pg ] .(5) Under the hypotheses as in Theorem 4.19(b), ( p, g ) is a regular sequence on B .Proof. (1): This follows from the combination of Lemma 3.4 and Lemma 3.7.(2): By our assumption combined with Lemma 3.7(2), the map A [ p j g ] → B [ p j g ] is integral andbecomes finite ´etale after inverting p . One easily checks that p A j ◦ ⊂ ( pg ) pn A j ◦ for any n > p A j ◦ ⊂ [ A [ p j g ] in view of (4 . p ( A [ p j g ]) + A [ pg ] ⊂ A [ p j g ] in view of [44, Proposition2.3]. Now we can apply [44, Proposition 4.5(4)] and deduce that (cid:0) B [ p j g ] , ( p ) (cid:1) is preuniform. Since (cid:0) B [ p j g ] (cid:1) + B ′ = (cid:0) A [ p j g ] (cid:1) + B ′ and (cid:0) ( A [ p j g ]) + B ′ , ( p ) (cid:1) is a pair of definition of B j by [44, Corollary 4.8(1)],the assertion follows.(3): We denote by b B , [ A [ p j g ] and [ B [ p j g ] the p -adic completions, respectively. Let B j be the imageof the natural map B [ p j g ] ⊗ A [ pjg ] [ A [ p j g ] → B j . Then by the assertion (1) and [44, Lemma 2.20], wesee that ( B j , ( p )) is a pair of definition of B j . Moreover by [44, Theorem 4.9(1)], the natural map B [ p j g ] ⊗ A [ pjg ] [ A [ p j g ] → [ B [ p j g ] induces an isomorphism(4.21) B j ∼ = −→ \ B [ p j g ] . Inverting p in (4.21), we obtain the desired isomorphism of topological rings B j ∼ = −→ B{ p j g } .(4): Since B is integral over A and p ∈ A is contained in the Jacobson radical, p ∈ B is alsocontained in the Jacobson radical of B . Since B is integral over a Noetherian ring and integrallyclosed in B ′ = B [ pg ], it is also completely integrally closed in B ′ by Proposition 6.1.(5): By assumption, A is torsion free and integral over some Noetherian normal domain R , thefield of fractions of R has characteristic zero, A is normal, and A [ pg ] → B ′ is finite ´etale. Thesefacts combine together to show that B ′ is a normal ring and B is the filtered colimit of normal rings that are torsion free and module-finite over R . Thus, ( p, g ) forms a regular sequence on B inview of Serre’s normality criterion. (cid:3) Theorem 4.19 (Witt-perfect Abhyankar’s lemma) . Let A be a p -torsion free ring that is flat over aWitt-perfect valuation domain V of rank admitting a compatible system of p -power roots p pn ∈ V ,together with a nonzero divisor g ∈ A admitting a compatible system of p -power roots g pn ∈ A .Suppose that(1) A is p -adically Zariskian and A is completely integrally closed in A [ pg ] ;(2) A is a ( pg ) p ∞ -almost Witt-perfect ring that is integral over a Noetherian ring;(3) ( p, g ) is a regular sequence on A (or more generally, it suffices to assume that p, g arenonzero divisors on the p -adic completion b A );(4) A [ pg ] ֒ → B ′ is a finite ´etale extension.Let us put g − p ∞ A := n a ∈ A [ 1 g ] (cid:12)(cid:12)(cid:12) g pn a ∈ A, ∀ n > o , which is an A -subalgebra of A [ g ] . Denote by B := ( g − p ∞ A ) + B ′ the integral closure of g − p ∞ A in B ′ (which is equal to the integral closure of A in B ′ by Lemma 4.18). Then the followingstatements hold:(a) The Frobenius endomorphism F rob : B/ ( p ) → B/ ( p ) is ( pg ) p ∞ -almost surjective andit induces an injection B/ ( p p ) ֒ → B/ ( p ) .(b) Suppose that A is a normal ring that is torsion free and integral over a Noetheriannormal domain. Then the induced map A/ ( p m ) → B/ ( p m ) is ( pg ) p ∞ -almost finite´etale for all m > .Proof. That B is the integral closure A in B ′ was already proved in Lemma 4.18. In order to provethe theorem, we use Galois theory of commutative rings. By decomposing A into the direct productof rings, we may assume and do that A [ pg ] → B ′ is finite ´etale of constant rank (indeed, one cancheck the conditions (1) ∼ (4) remain to hold for each direct factor of the ring A ). By [1, Lemme1.9.2] applied to the finite ´etale extension A [ pg ] ֒ → B ′ = B [ pg ], there is the decomposition(4.22) A [ 1 pg ] ֒ → B ′ = B [ 1 pg ] ֒ → C ′ , where A [ pg ] → C ′ and B ′ = B [ pg ] → C ′ are Galois coverings and let G be the Galois group for A [ pg ] → C ′ . Let B j (resp. C j ) be the resulting Tate ring according to Lemma 4.18.We shall fix the notation: A , K and A j as defined in (4 .
5) and (4 . K is a perfectoid field, A is a ( pg ) p ∞ -almost perfectoid and A j are perfectoid K -algebras. Considerthe complete and separated Tate ring: B j := B ′ ⊗ A [ pg ] A j (resp. C j := C ′ ⊗ A [ pg ] A j ) as in Lemma4.18. Then one can equip B j (resp. C j ) with a norm assoiated to a pair of definition of it so that B j (resp. C j ) is a Banach A j -algebra. Since A [ pg ] → B ′ (resp. A [ pg ] → C ′ ) is finite ´etale, A j → B j (resp. A j → C j ) is also finite ´etale. By [50, Theorem 7.9], B j (resp. C j ) is a perfectoid K -algebra. VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 23
Moreover, we have a natural commutative diagram B ′ ψ j +1 −−−−→ B j +1 (cid:13)(cid:13)(cid:13) y B ′ ψ j −−−−→ B j and the set of A -algebras {B j } j> forms an inverse system, where B j +1 → B j is the natural inclusiondefined by the rule (4 . B j ◦ = ψ − j ( B j ◦ ) by Theorem 4.15(b) and [44, Corollary 4.10], weobtain the following commutative diagram: A −−−−→ A j +1 ◦ −−−−→ B j +1 ◦ −−−−→ B ′ (cid:13)(cid:13)(cid:13) y y (cid:13)(cid:13)(cid:13) A −−−−→ A j ◦ −−−−→ B j ◦ −−−−→ B ′ Taking inverse limits, we have compositions of ring maps:(4.23) A ∼ = e A := lim ←− j A j ◦ → e B := lim ←− j B j ◦ → B ′ , where the first isomorphism is due to Theorem 4.15. Similarly, after setting(4.24) e C := lim ←− j C j ◦ , we obtain the compositions of ring maps A ∼ = e A → e C → C ′ .Let us prove the assertion ( a ). By Lemma 3.9, the action of G preserves C j ◦ and we have( C j ◦ ) G = A j ◦ . Hence A j ◦ → C j ◦ is an integral extension. Since B j → C j carries B j ◦ into C j ◦ by[44, Lemma 2.18], A j ◦ → B j ◦ is also integral. Taking G -invariants of rings appearing in (4 . e C G ∼ = (lim ←− j C j ◦ ) G ∼ = lim ←− j ( C j ◦ ) G ∼ = lim ←− j A j ◦ ∼ = e A, which implies that e A → e C is integral. Hence e A → e B is integral, fitting into the commutativesquare:(4.25) e A −−−−→ e B (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) A −−−−→ B Since A j ◦ [ p ] → B j ◦ [ p ] is identified with the finite ´etale extension A [ pg ] → B ′ = B [ pg ] and B j ◦ is the integral closure of A j ◦ in B ′ , it follows from the almost purity theorem for Witt-perfectrings [16, Theorem 5.2] or [17, Theorem 2.9], that A j ◦ → B j ◦ is ( p ) p ∞ -almost finite ´etale. Inparticular, B j ◦ is a Witt-perfect V -algebra. Retain the notation as in (4 . A ∼ = g − p ∞ A ∼ = e A . Since A is ( pg ) p ∞ -almost Witt-perfect by assumption, the p -adic completion be A is an integral ( pg ) p ∞ -almost perfectoid ring. Then we claim that(4.26) be B is integral ( pg ) p ∞ -almost perfectoid . In [1, Question 3.5.1], a question is raised as to whether g − p ∞ b A [ p ] ◦ is integral perfectoid. By applying [1, Proposition 4.4.1], for any fixed r = np with n ∈ N , we get a ( pg ) p ∞ -almostisomorphism:(4.27) lim ←− j (cid:0) B j ◦ / ( p r ) (cid:1) ≈ . After applying lim ←− to the short exact sequence 0 → B j ◦ / ( p p − p ) → B j ◦ / ( p ) → B j ◦ / ( p p ) →
0, thefollowing ( pg ) p ∞ -almost surjection follows from (4 . ←− j B j ◦ / ( p ) → lim ←− j B j ◦ / ( p p ) . By Witt-perfectness of B j ◦ , the Frobenius isomorphism B j ◦ / ( p p ) ∼ = B j ◦ / ( p ) yields that(4.29) lim ←− j (cid:0) B j ◦ / ( p ) (cid:1) F rob −−−→ lim ←− j (cid:0) B j ◦ / ( p ) (cid:1) is ( pg ) p ∞ -almost surjective . Consider the commutative diagramlim ←− j (cid:0) B j ◦ / ( p ) (cid:1) F rob −−−−→ lim ←− j (cid:0) B j ◦ / ( p ) (cid:1)x x(cid:0) lim ←− j B j ◦ (cid:1) / ( p ) F rob −−−−→ (cid:0) lim ←− j B j ◦ (cid:1) / ( p )In order to prove (4 . (cid:0) lim ←− j B j ◦ (cid:1) / ( p ) → lim ←− j (cid:0) B j ◦ / ( p ) (cid:1) is a ( pg ) p ∞ -almost isomorphism in view of (4 . pg ) p ∞ -almost surjective by applying the almost surjectivity of (4 .
28) to [1,Proposition 4.3.1 and Remarque 4.3.1] and thus, the Frobenius endomorphism on e B/ ( p ) is ( pg ) p ∞ -almost surjective. Notice that the diagram (4 .
25) implies that B = e B . So the assertion ( a ) isproved.Finally we prove ( b ) and fix the notation as in ( a ). Then A [ pg ] → C ′ is a G -Galois covering, A j → C j is also a G -Galois covering by [19, Lemma 12.2.7]. Let d C j ◦ be the p -adic completion of C j ◦ . Since C j ◦ [ p ] = C ′ , there is a natural A j -algebra homomorphism(4.30) C j = C ′ ⊗ A [ pg ] A j → d C j ◦ [ 1 p ] . Since d A j ◦ ∼ = A j ◦ by Theorem 4.15(b), the map (4.30) is an isomorphism, which induces C j ◦ ∼ = d C j ◦ in view of [44, Corollary 4.10]. Thus, G acts on d C j ◦ and(4.31) ( d C j ◦ ) G ∼ = ( C j ◦ ) G ∼ = A j ◦ by applying Lemma 3.9 or Discussion 4.20(1) below. In particular, A j ◦ → d C j ◦ is an integralextension. In summary,(4.32) A j ◦ → C j ◦ ∼ = d C j ◦ is ( p ) p ∞ -almost ´etale and A j → C j ∼ = d C j ◦ [ 1 p ] is Galois with Galois group G. To finish the proof, let us apply the proof of [1, Proposition 5.2.3] via Galois theory of commuta-tive rings to (4 . VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 25 for the following discussions. Let us set C := (lim ←− j d C j ◦ )[ 1 p ]and in particular, C ◦ ∼ = lim ←− j C j ◦ . After invoking (4 . C ◦ ∼ = be C or equivalently , \ lim ←− j C j ◦ ∼ = lim ←− j d C j ◦ . Hence(4.34) ( be C ) G ∼ = (cid:0) \ lim ←− j C j ◦ (cid:1) G ∼ = (cid:0) lim ←− j d C j ◦ (cid:1) G ∼ = lim ←− j ( C j ◦ ) G ∼ = lim ←− j A j ◦ ∼ = lim ←− j d A j ◦ ∼ = be A, where the third isomorphism follows from the commutativity of inverse limits with taking G -invariants and (4 . . .
32) and applying [1, Proposition 3.3.4], the map(4.35) C j ◦ b ⊗ A j ◦ C j ◦ → Y G C j ◦ defined by b ⊗ b ′ (cid:0) γ ( b ) b ′ (cid:1) γ ∈ G is a ( p ) p ∞ -almost isomorphism, where the completed tensor product is p -adic. By [1, Proposition4.4.4], we have C{ p j g } ∼ = C j and C is an A -algebra. Using this, we obtain (cid:0) C b ⊗ A C (cid:1) { p j g } ∼ = C b ⊗ A C b ⊗ A A j ∼ = (cid:0) C b ⊗ A A j (cid:1) ⊗ A j (cid:0) C b ⊗ A A j (cid:1) ∼ = C{ p j g } ⊗ A j C{ p j g } ∼ = C j ⊗ A j C j . By Riemann’s extension theorem [1, Th´eor`eme 4.2.2] and by [1, Proposition 3.3.4], we have ( pg ) p ∞ -almost isomorphisms:(4.36) lim ←− j (cid:0) C j ◦ b ⊗ A j ◦ C j ◦ (cid:1) ≈ lim ←− j (cid:0) C j ⊗ A j C j (cid:1) ◦ ∼ = lim ←− j (cid:0) C b ⊗ A C (cid:1) { p j g } ◦ ≈ (cid:0) C b ⊗ A C (cid:1) ◦ ≈ C ◦ b ⊗ A ◦ C ◦ . Putting (4 .
35) and (4 .
36) together, we get the following ( pg ) p ∞ -almost isomorphism:(4.37) C ◦ b ⊗ A ◦ C ◦ ≈ Y G C ◦ . After making reductions of (4 .
34) and (4 .
37) modulo p m for m > . A ◦ / ( p m ) → (cid:0) C ◦ / ( p m ) (cid:1) G is a ( pg ) p ∞ -almost isomorphism via Discussion 4.20(2). So theinduced map: e A/ ( p m ) → e C/ ( p m ) is a ( pg ) p ∞ -almost G -Galois covering. This map factors as e A/ ( p m ) → e B/ ( p m ) → e C/ ( p m ). It then follows from [1, Proposition 1.9.1(3)] that e A/ ( p m ) → e B/ ( p m )is ( pg ) p ∞ -almost finite ´etale, as desired. This completes the proof of the theorem. (cid:3) Discussion 4.20. (1) Here is a way to check the isomorphism: ( d C j ◦ ) G ∼ = A j ◦ that appears in(4 . G -invariants and d A j ◦ ∼ = A j ◦ by Theorem4.15, we have(4.38)( d C j ◦ ) G ∼ = (cid:0) lim ←− m C j ◦ / ( p m ) (cid:1) G ∼ = lim ←− m (cid:0) C j ◦ / ( p m ) (cid:1) G ≈ lim ←− m (cid:0) ( C j ◦ ) G / ( p m ) (cid:1) ∼ = lim ←− m A j ◦ / ( p m ) ∼ = A j ◦ , where ≈ in the middle denotes a ( p ) p ∞ -almost isomorphism and we reason this as follows:Consider the short exact sequence 0 → C j ◦ p m −−→ C j ◦ → C j ◦ / ( p m ) →
0. Applying theGalois cohomology H i ( G, ) to this exact sequence, we get an injection ( C j ◦ ) G / ( p m ) ֒ → (cid:0) C j ◦ / ( p m ) (cid:1) G whose cokernel embeds into H ( G, C j ◦ ). By applying [18, Theorem 2.4] or[45, Proposition 3.4], H ( G, C j ◦ ) is ( p p ∞ )-almost zero. Hence (4 .
38) is proved. d C j ◦ iscompletely integrally closed in d C j ◦ [ p ] by Lemma 3.3. Then we have d C j ◦ ∼ = p − p ∞ ( d C j ◦ )and p − p ∞ ( A j ◦ ) ∼ = A j ◦ by Lemma 3.4. Since the functor p − p ∞ ( ) commutes with taking G -invariants, (4 .
38) yields an (honest) isomorphism:( d C j ◦ ) G ∼ = (cid:0) p − p ∞ ( d C j ◦ ) (cid:1) G ∼ = p − p ∞ (cid:0) ( d C j ◦ ) G (cid:1) ∼ = p − p ∞ ( A j ◦ ) ∼ = A j ◦ , which proves (4 . A ◦ / ( p m ) → (cid:0) C ◦ / ( p m ) (cid:1) G is a ( pg ) p ∞ -almost isomorphism for any integer m >
0. We have already seen the ( pg ) p ∞ -almost isomorphisms: A j ◦ / ( p m ) ≈ ( C j ◦ ) G / ( p m ) ≈ (cid:0) C j ◦ / ( p m ) (cid:1) G . Taking the inverse limits j → ∞ and using [1, Proposition 4.2.1], we get ( pg ) p ∞ -almost isomorphisms: A ◦ / ( p m ) ≈ lim ←− j> (cid:0) C j ◦ / ( p m ) (cid:1) G ∼ = (cid:0) lim ←− j> C j ◦ / ( p m ) (cid:1) G ≈ (cid:0) C ◦ / ( p m ) (cid:1) G , as wanted. Problem 3.
Does Theorem 4.19 hold true under the more general assumption that A is not nec-essarily integral over a Noetherian ring? This problem is related to a possible generalization of Riemann’s extension theorem (see Theorem4.15) for Witt-perfect rings of general type.5.
Applications of Witt-perfectoid Abhyankar’s lemma
Construction of almost Cohen-Macaulay algebras.
Before proving the main theoremfor this section, we recall the definition of big Cohen-Macaulay algebras, due to Hochster.
Definition 5.1 (Big Cohen-Macaulay algebra) . Let ( R, m ) be a Noetherian local ring of dimension d > T be an R -algebra. Then T is a big Cohen-Macaulay R -algebra , if there is a systemof parameters x , . . . , x d such that x , . . . , x d is a regular sequence on T and ( x , . . . , x d ) T = T .Moreover, we say that a big Cohen-Macaulay algebra is balanced , if every system of parameterssatisfies the above conditions.We also recall the definition of almost Cohen-Macaulay algebras from [2, Definition 4.1.1]. Referthe reader to [3, Proposition 2.5.1] for a subtle point on this definition. Definition 5.2 (Almost Cohen-Macaulay algebra) . Let ( R, m ) be a Noetherian local ring of di-mension d >
0, and let (
T, I ) be a basic setup equipped with an R -algebra structure. Fix a systemof parameters x , . . . , x d . We say that T is I -almost Cohen-Macaulay with respect to x , . . . , x d , if T / m T is not I -almost zero and c · (cid:0) ( x , . . . , x i ) : T x i +1 (cid:1) ⊂ ( x , . . . , x i ) T for any c ∈ I and i = 0 , . . . , d − VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 27
It is important to keep in mind that the permutation of the sequence x , . . . , x d in the abovedefinition may fail to form an almost regular sequence. We consider the sequence p, x , . . . , x d forthe main theorem below. Andr´e’s construction : For the applications given below, we take I to be the ideal S n> π pn T as the basic setup ( T, I ) for some nonzero divisor π ∈ R . Following [2], we introduce some auxiliaryalgebras. Let W ( k ) be the ring of Witt vectors for a perfect field k of characteristic p > A := W ( k )[[ x , . . . , x d ]]be an unramified complete regular local ring and V j := W ( k )[ p pj ]. Then V j is a complete discretevaluation ring and set V ∞ := lim −→ j V j . Then this is a Witt-perfect valuation domain. For a fixedelement 0 = g ∈ A , we set B jk := V j [[ x pj , . . . , x pj d ]][ g pk ][ 1 p ] := (cid:16) V j [[ x pj , . . . , x pj d ]][ T ] / ( T p k − g ) (cid:17) [ 1 p ]for any pair of non-negative integers ( j, k ). For any pairs ( j, k ) and ( j ′ , k ′ ) with j ≤ j ′ and k ≤ k ′ ,we can define the natural map B jk → B j ′ k ′ . Let us define the A -algebra A jk to be the integralclosure of A in B jk . Let us also define(5.1) A ∞∞ := lim −→ j,k A jk and A ∞ g := the integral closure of A ∞∞ in A ∞∞ [ 1 pg ] . For brevity, let us write(5.2) A ∞ := A ∞ := lim −→ j V j [[ x pj , . . . , x pj d ]] . Then we have towers of integral ring maps: A → A ∞ → A ∞∞ → A ∞ g . Lemma 5.3.
Let R be a Noetherian domain with a proper ideal I and let T be a normal ring that isa torsion free integral extension of R . Assume that π ∈ I is a nonzero element such that T admitsa compatible system of p -power roots π pn . Then T /IT is not ( π p ∞ ) -almost zero.Proof. In order to prove that
T /IT is not ( π ) p ∞ -almost zero, it suffices to prove that T m /IT m is not ( π ) p ∞ -almost zero, where m is any maximal ideal of T containing IT , since T m /IT m is thelocalization of T /IT . Then T m is a normal domain that is an integral extension over the Noetheriandomain R m ∩ R , in which I is a proper ideal. To derive a contradiction, we suppose that T m /IT m is ( π p ∞ )-almost zero. Notice that T m is contained in the absolute integral closure ( R m ∩ R ) + . Inparticular, it implies that ( π ) pn ∈ IT m for all n > . Raising p n -th power on both sides, we get by [52, Lemma 4.2]; π ∈ \ n> I p n T m = 0 , which is a contradiction. (cid:3) Proposition 5.4.
Let the notation be as in (5 . and (5 . . Then the following assertions hold: (1) A ∞ is completely integrally closed in its field of fractions that is an integral and faithfullyflat extension over A . Moreover, the localization map A ∞ [ pg ] → A ∞∞ [ pg ] is ind-´etale.(2) A ∞ g is a ( g ) p ∞ -almost Witt-perfect algebra over the Witt-perfect valuation domain V ∞ suchthat p pn ∈ V ∞ , g pn ∈ A ∞ g . Moreover, A ∞ g is a ( pg ) p ∞ -almost Cohen-Macaulay normalring that is completely integrally closed in A ∞ g [ pg ] . In particular, the localization of A ∞ g at its any maximal ideal is a ( pg ) p ∞ -almost Cohen-Macaulay normal domain.Proof. (1): It is clear that A → A ∞ is integral by construction. Since A ∞ is a filtered colimitof regular local subrings with module-finite transition maps, one readily checks that A → A ∞ isfaithfully flat. By Lemma 3.7, A ∞ is a completely integrally closed domain in its field of fractions.By looking at the discriminant, it is easy to check that A ∞ [ pg ] → A ∞∞ [ pg ] is ind-´etale.(2): By Andr´e’s crucial result [2, Th´eor`eme 2.5.2] combined with Lemma 5.3, we find that A ∞∞ is a ( p ) p ∞ -almost Cohen-Macaulay and Witt-perfect algebra. Next let us study A ∞ g and consider e A ∞∞ := lim ←− j A j ◦∞∞ attached to A ∞∞ as defined in Theorem 4.15. Then we claim that(5.3) A ∞ g ∼ = e A ∞∞ . Notice that since A ∞ g is integrally closed in A ∞∞ [ pg ] = A ∞ g [ pg ], it follows from Proposition 6.1that A ∞ g is completely integrally closed in A ∞∞ [ pg ].Now by applying Theorem 4.15(c) to A ∞∞ , the equality (5 .
3) follows, where one should noticethat p is in the Jacobson radical of A ∞∞ and g remains a nonzero divisor on the p -adic completion b A ∞∞ in view of [2, Remarques 2.6.1]. It follows from Riemann’s extension theorem [1, Th´eor`eme4.2.2] combined with (5 .
3) that g − p ∞ b A ∞∞ ∼ = lim ←− j \ A j ◦∞∞ ∼ = \ lim ←− j A j ◦∞∞ ∼ = b A ∞ g , where the middle isomorphism is due to Proposition 4.16. In particular, b A ∞∞ → b A ∞ g is a ( g ) p ∞ -almost isomorphism.From the property of A ∞∞ mentioned in (1), one finds that b A ∞ g is an integral ( g ) p ∞ -almostperfectoid and ( pg ) p ∞ -almost Cohen-Macaulay algebra. By the fact that A ∞ [ pg ] is a normal domainand A ∞ [ pg ] → A ∞∞ [ pg ] is obtained as a filtered colimit of finite ´etale A ∞ [ pg ]-algebras, we see that A ∞∞ [ pg ] is a normal ring; the localization at any maximal ideal is an integrally closed domain byLemma 3.8. Since A ∞ g is integrally closed in A ∞∞ [ pg ], it follows that A ∞ g is also normal. (cid:3) As a corollary, we obtain the following theorem.
Theorem 5.5.
Let ( R, m ) be a complete Noetherian local domain of mixed characteristic p > with perfect residue field k . Let p, x , . . . , x d be a system of parameters and let R + be the absoluteintegral closure of R . Then there exists an R -algebra T together with a nonzero element g ∈ R suchthat the following hold:(1) T admits compatible systems of p -power roots p pn , g pn ∈ T for all n > .(2) The Frobenius endomorphism F rob : T / ( p ) → T / ( p ) is surjective. A similar construction also appears in [22, Theorem 16.9.17], where they apply p -integral closure instead ofintegral closure. This makes it possible to get rid of ”( p ) p ∞ -almost” from the statement. VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 29 (3) T is a ( pg ) p ∞ -almost Cohen-Macaulay normal domain with respect to p, x , . . . , x d and R ⊂ T ⊂ R + .(4) R [ pg ] → T [ pg ] is an ind-´etale extension. In other words, T [ pg ] is a filtered colimit of finite´etale R [ pg ] -algebras contained in T [ pg ] .Proof. In the following, we may assume dim R ≥ A := W ( k )[[ x , . . . , x d ]] ֒ → R. As the induced field extension Frac( A ) → Frac( R ) is separable, there is an element g ∈ A \ pA suchthat A [ pg ] → R [ pg ] is ´etale. As in Proposition 5.4, we set A ∞ := [ n> W ( k )[ p pn ][[ x pn , . . . , x pn d ]] . Now consider the integral extensions A → A ∞ → A ∞∞ → A ∞ g as in Proposition 5.4. Let n bea maximal ideal of A ∞ g . Then the localization ( A ∞ g ) n is a normal domain that is an integralextension over A and enjoys the same properties as A ∞ g . Since ( p, g ) forms part of a system ofparameters of A and ( A ∞ g ) n is a filtered colimit of module-finite normal A -algebras, it follows that( p, g ) is a regular sequence on ( A ∞ g ) n by Serre’s normality criterion. By base change, the map(5.4) ( A ∞ g ) n [ 1 pg ] → R ⊗ A ( A ∞ g ) n [ 1 pg ]is finite ´etale. Then R ⊗ A ( A ∞ g ) n [ pg ] is a normal ring. Letting the notation be as in (5 . B := the integral closure of R in R ⊗ A ( A ∞ g ) n [ 1 pg ] . Then by the normality of R ⊗ A ( A ∞ g ) n [ pg ] and Lemma 3.8, it follows that B is a normal ring thatfits into the commutative diagram: ( A ∞ g ) n −−−−→ B x x A −−−−→ R in which every map is injective and integral. Let n ′ be any maximal ideal of B . Since A is a localdomain and A → B is a torsion free integral extension, one finds that A ∩ n ′ is the unique maximalideal of A and the induced localization map A → B n ′ is an injective integral extension betweennormal domains. By setting A := ( A ∞ g ) n in the notation of Theorem 4.19 and applying Lemma 5.3,it follows that B is a ( pg ) p ∞ -almost Cohen-Macaulay normal ring with respect to p, x , . . . , x d and( pg ) p ∞ -almost Witt-perfect. Since these properties are preserved under localization with respectto any maximal ideal, it follows that the normal domain B n ′ enjoys the same properties.To finish the proof, let us put C := B n ′ for brevity of notation. Set T := the integral closure of C in C [ 1 p ] ´et , where C [ p ] ´et is the maximal ´etale extension of C [ p ] contained in the absolute integral closure C [ p ] + .Then T is a Witt-perfect normal domain in view of [55, Lemma 5.1] or [56, Lemma 10.1]. Therefore, In what follows, if necessary, we repeat the same argument for deriving the regularity of ( p, g ) in order to applyTheorem 4.19. it remains to establish that T is ( pg ) p ∞ -almost Cohen-Macaulay with respect to p, x , . . . , x d . Letus note that the composite map ( A ∞ g ) n [ 1 pg ] → C [ 1 pg ] → T [ 1 pg ]is an ind-´etale extension. So we find that T [ pg ] is the filtered colimit of finite ´etale ( A ∞ g ) n [ pg ]-algebras. As T is integrally closed in its field of fractions, the integral closure of ( A ∞ g ) n in T [ pg ]is the same as T . Summing up, we conclude from Theorem 4.19 applied to A := ( A ∞ g ) n , togetherwith the fact that A ∞∞ / ( p ) → A ∞ g / ( p ) is a ( g ) p ∞ -almost isomorphism, that T / ( p ) is the filteredcolimit of ( pg ) p ∞ -almost finite ´etale A ∞∞ / ( p )-algebras. By Lemma 5.3 and Proposition 5.4, T is( pg ) p ∞ -almost Cohen-Macaulay. (cid:3) As a corollary, we obtain the following result, which is the strengthened version of the mainresults in [29]. The proof uses standard results from the theory of local cohomology.
Corollary 5.6.
Let the notation and hypotheses be as in Theorem 5.5. Then the local cohomologymodules H i m ( T ) are ( pg ) p ∞ -almost zero in the range ≤ i ≤ dim R − . In particular, the imageof the map H i m ( T ) → H i m ( R + ) induced by T → R + is ( pg ) p ∞ -almost zero.Proof. Letting p, x , . . . , x d be a system of parameters of R , if one inspects the structure of theproof of Theorem 5.5 and Theorem 4.19, it follows that x m , . . . , x md forms a ( pg ) p ∞ -almost regularsequence on T / ( p m ) for all integers m >
0. As in the proof of [29, Theorem 3.17], the Koszulcohomology modules H i ( p m , x m , . . . , x md ; T ) and hence H i m ( T ) are ( pg ) p ∞ -almost zero for i < dim R . (cid:3) It is reasonable to study the following problem, which we credit to Heitmann in the 3-dimensionalcase thanks to his proof of the direct summand conjecture; see [27].
Problem 4.
Let ( R, m ) be a complete Noetherian local domain of arbitrary characteristic withits absolute integral closure R + and the unique maximal ideal m R + . Fix a system of parameters x , . . . , x d of R . Then does it hold true that c · (cid:0) ( x , . . . , x i ) : R + x i +1 (cid:1) ⊂ ( x , . . . , x i ) R + for any c ∈ m R + and i = 0 , . . . , d − ? Bhatt gave an answer to the above problem in mixed characteristic in [10] when x = p n , andHochster and Huneke gave a complete answer in the characteristic p > arbitrary system of parameters. Itseems that perfectoids or other similar techniques do not suffice to study the problem. Problem 5.
Let T be a big Cohen-Macaulay algebra over a Noetherian local domain ( R, m ) ofmixed characteristic. Then does T map to an integral perfectoid big Cohen-Macaulay R -algebra? Here we mention a few related results.
Proposition 5.7.
Assume that T is a big Cohen-Macaulay algebra over a Noetherian local domain ( R, m ) of any characteristic. Then T maps to an R -algebra B such that the following hold:(1) B is free over T . In particular, B is a big Cohen-Macaulay R -algebra. VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 31 (2) B is absolutely integrally closed. In other words, every nonzero monic polynomial in B [ X ] has a root in B .Proof. Just apply [57, Tag 0DCR]. (cid:3)
Theorem 5.8 (Gabber-Ramero) . Let ( R, m ) be a complete local domain of mixed characteristic.Then any integral perfectoid big Cohen-Macaulay R -algebra B admits an R -algebra map B → C such that C is an integral perfectoid big Cohen-Macaulay R -algebra and C is an absolutely integrallyclosed quasi-local domain.Proof. See [22, Theorem 17.5.96]. (cid:3)
Problem 6.
Let ( R, m ) be a complete Noetherian local domain of mixed characteristic. Then canone construct a big Cohen-Macaulay R -algebra T such that T has bounded p -power roots of p orequivalently, the radical ideal √ pT is finitely generated? So far, big Cohen-Macaulay algebras constructed using perfectoids necessarily admit p -powerroots of p and we do not know if the construction as stated in the problem is possible.5.2. Construction of big Cohen-Macaulay modules.
We demonstrate a method of construct-ing a big Cohen-Macaulay module by using the R -algebra T from Theorem 5.5. Corollary 5.9.
Let the notation be as in Theorem 5.5. Set M := ( pg ) p ∞ T . Then M is an idealof T that is ( pg ) p ∞ -almost isomorphic to T , and M is a big Cohen-Macaulay R -module. In otherwords, H i m ( M ) = 0 for all ≤ i ≤ dim R − .Proof. Notice that T is pg -torsion free and there is an isomorphism as T -modules: T ∼ = ( pg ) pn T .Consider the commutative diagram: T × ( pg ) − p −−−−−−→ T × ( pg ) p − p −−−−−−−→ T −−−−→ · · · × pg y × ( pg ) p y × ( pg ) p y ( pg ) T −−−−→ ( pg ) p T −−−−→ ( pg ) p T −−−−→ · · · where the horizontal arrows in the bottom are natural injections, and the vertical arrows arebijections. Fix any i < dim R . Applying the local cohomology to this commutative diagram, thebottom horizontal sequence becomes:(5.5) lim −→ n> H i m (( pg ) pn T ) ∼ = H i m (lim −→ n> ( pg ) pn T ) ∼ = H i m ( M ) , where the first isomorphism uses the commutativity of cohomolgical functor with direct limit. Thehorizontal upper sequence becomes:(5.6) lim −→ n n H i m ( T ) × ( pg ) − p −−−−−−→ H i m ( T ) × ( pg ) p − p −−−−−−−→ H i m ( T ) → · · · o ∼ = 0 , because the local cohomology modules H i m ( T ) are annihilated by ( pg ) pn for any n > i < dim R . As (5 .
5) and (5 .
6) yield the isomorphic modules, we have the desired vanishing cohomology.Since R is a Noetherian local domain, there is a discrete valuation v : R → Z ≥ ∪ {∞} withcenter on the maximal ideal. Then one extends v as a Q -valued valuation on T . One can usethis valuation to deduce that M = m M and the details are left as an exercise; see also [6, Lemma3.15]. (cid:3) Appendix: Integrality and almost integrality
In this appendix, our aim is to give a proof to the following result (see Proposition 6.1). Forgeneralities on topological spaces and maximal separated quotients , we refer the reader to [20,Chapter 0, 2.3(c)] and [39, Definition 2.4.8]. For a topological space X , we denote by [ X ] themaximal separated quotient of X , thus defining the natural epimorphism X → [ X ]. Proposition 6.1.
Let A be a ring that is integral over a Noetherian ring, and let t ∈ A be anonzero divisor. Then an element a ∈ A [ t ] is integral over A if and only if it is almost integralover A . More precisely, for the Tate ring A associated to ( A , ( t )) , we have ( A ) + A = A ◦ = n a ∈ A (cid:12)(cid:12)(cid:12) | a | x ≤ for any x ∈ [Spa( A, ( A ) + A )] o . The idea of our proof is to reduce the assertion to the situation of Corollary 4.12, using
Zariskiza-tion . Let us recall its definition below (see also [20, Chapter 0, 7.3(b)] or [59, Definition 3.1]).
Definition 6.2.
Let A be a ring and I ⊂ A be an ideal. Then we denote by A ZarI the localization(1 + I ) − A , and call it the I -adic Zariskization of A .We will utilize the following properties of Zariskization. Lemma 6.3.
Let A ⊂ B be an integral ring extension and let I ⊂ A be an ideal. Then the followingassertions hold.(1) The induced ring map A ZarI → B ZarIB is also integral.(2) Let { A λ } λ ∈ Λ be the filtered system of all module-finite A -subalgebras of B . Then we have acanonical isomorphism of rings lim −→ λ ( A λ ) ZarIA λ ∼ = −→ B ZarIB .Proof. (1): Set B ′ = B ⊗ A A ZarI . Then the map A ZarI → B ZarIB is given as the composite of theintegral map A ZarI → B ′ and the canonical B -algebra homomorphism B ′ → B ZarIB . Moreover,since B ′ is IB ′ -adically Zariskian, we have the B -algebra homomorphism B ZarIB → B ′ . Since thecomposite B ZarIB → B ′ → B ZarIB is the identity map by the universal property, the map B ′ → B ZarIB is surjective. Hence the assertion follows.(2): Since B is integral over A , we have lim −→ λ A λ = B . For each λ ∈ Λ, the map A λ ֒ → B inducesthe A λ -algebra homomorphism ϕ λ : ( A λ ) ZarIA λ → B ZarIB . Hence we have the B -algebra homomorphism ϕ : lim −→ λ ( A λ ) ZarIA λ → B ZarIB . Now for any x ∈ IB , there exists some λ ∈ Λ such that 1 + x ∈ IA λ .Hence ϕ is injective. Set C := lim −→ λ ( A λ ) ZarIA λ . Then, since A ZarI → C is integral by the assertion (1), C is IC -adically Zariskian. Hence we obtain the B -algebra homomorphism ψ : B ZarIB → C ZarIC , andthe composite ϕ ◦ ψ is the identity map by the universal property. Therefore ϕ is surjective. Thusthe assertion follows. (cid:3) Corollary 6.4.
Let A be a ring with a nonzero divisor t ∈ A . Put A := A [ t ] and A ′ :=( A ) Zar ( t ) [ t ] . Then the inclusion A ֒ → ( A ) + A induces an isomorphism (( A ) Zar ( t ) ) + A ′ ∼ = −→ (( A ) + A ) Zar ( t ) .Proof. Since integrality of a ring extension is preserved under localization, it suffices to show that(( A ) + A ) Zar ( t ) ∼ = ( A ) + A ⊗ A ( A ) Zar ( t ) . First, we have an isomorphism lim −→ λ ( A λ ) Zar ( t ) ∼ = −→ (( A ) + A ) Zar ( t ) byLemma 6.3(2). Moreover for each λ ∈ Λ, there exists some m > t m A λ ⊂ A . Then,since 1 + t m +1 A λ ⊂ tA , we have ( A λ ) Zar ( t ) ∼ = ( A λ ) Zar ( t m +1 ) ∼ = A λ ⊗ A ( A ) Zar ( t ) . Thus the assertionfollows. (cid:3) Now we can complete the proof of Proposition 6.1.
VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN-MACAULAY ALGEBRAS 33
Proof of Proposition 6.1.
Set X = Spa( A, ( A ) + A ). Since we know that( A ) + A ⊂ A ◦ ⊂ n a ∈ A (cid:12)(cid:12)(cid:12) | a | x ≤ x ∈ [ X ] o , it suffices to show the reverse inclusion. Pick c ∈ A such that | c | x ≤ x ∈ [ X ]. Byassumption, there exists a Noetherian subring R ⊂ A such that t ∈ R and the filtered system { R λ } λ ∈ Λ of all module-finite R -subalgebras in A satisfies A = lim −→ λ R λ . Then by Lemma 6.3, A ′ := lim −→ λ ( R λ ) Zar ( t ) is integral over a Noetherian ring R Zar ( t ) . Let A ′ be the Tate ring associated to( A ′ , ( t )), and X ′ = Spa ( A ′ , ( A ′ ) + A ′ ). Then Corollary 4.12 implies that( A ′ ) + A ′ = ( A ′ ) ◦ = n a ∈ A ′ (cid:12)(cid:12)(cid:12) | a | x ′ ≤ x ′ ∈ [ X ′ ] o . Moreover, for the continuous ring map ψ : A → A ′ , we have | ψ ( c ) | x ′ ≤ x ′ ∈ X ′ byassumption. Thus we find that ψ ( c ) ∈ ( A ′ ) + A ′ . On the other hand, A ′ ∼ = ( A ) Zar ( t ) by Lemma 6.3and hence we have (( A ) + A ) Zar ( t ) ∼ = ( A ′ ) + A ′ by Lemma 6.4. Since the map ( A ) + A → (( A ) + A ) Zar ( t ) becomes an isomorphism after t -adic comple-tion, one can deduce from Beauville-Laszlo’s lemma (Lemma 3.2) that the diagram of ring maps( A ) + A / / (cid:15) (cid:15) (( A ) + A ) Zar ( t ) (cid:15) (cid:15) A ψ / / A ′ is cartesian. Thus we obtain c ∈ ( A ) + A , as wanted. (cid:3) Acknowledgement .
The authors are grateful to Professor K. Fujiwara for encouragement andcomments on this paper.
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