A VARIANT OF PERFECTOID ABHYANKAR’S LEMMA AND ALMOST COHEN–MACAULAY ALGEBRAS

Abstract In this article, we prove that a complete Noetherian local domain of mixed characteristic 
$p>0$
 with perfect residue field has an integral extension that is an integrally closed, almost Cohen–Macaulay domain such that the Frobenius map is surjective modulo p. This result is seen as a mixed characteristic analog of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen–Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André’s perfectoid Abhyankar’s lemma and Riemann’s extension theorem.


Introduction
In the present article, rings are assumed to be commutative with a unity. Recently, Yves André established Perfectoid Abhyankar's Lemma in [1] as a conceptual generalization of Almost Purity Theorem; see [57,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, André proved 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 by this result and led to the following commutative algebra question, which is raised in [53] and [54] implicitly.
Question 1 (Roberts). Let (R, m) be a complete Noetherian local domain of arbitrary characteristic with its absolute integral closure R + . Then does there exist an R-algebra B such that B is an almost Cohen-Macaulay R-algebra and R ⊂ B ⊂ R + ?
Essentially, Question 1 asks for a possibility to find a relatively small almost Cohen-Macaulay algebra. The structure of this article is twofold. We begin with giving an answer to Question 1 and then discuss necessary perfectoid techniques.
1.1. Main results on commutative algebra. André proved that any complete Noetherian local domain maps to a big Cohen-Macaulay algebra and using his result, it was proved that such a big Cohen-Macaulay algebra could be refined to be an integral perfectoid big Cohen-Macaulay algebra in [62]. See Definition 6.1 and Definition 6.2 for big (almost) Cohen-Macaulay algebras. Question 1 was stated in a characteristic-free manner. Let us point out that if dim R ≤ 2, then R + is a big Cohen-Macaulay algebra in an arbitrary characteristic. This is easily seen by using Serre's normality criterion. Recall that if R has prime characteristic p > 0, then R + is a big Cohen-Macaulay R-algebra. This result was proved by Hochster and Huneke and their proof is quite involved; see [32], [33], [34], [38], [52] and [55] for related results as well as [30], [31], [42], [47] and [48] for applications to tight closure, multiplier/test ideals and singularities on algebraic varieties. There is another important work on the purity of Brauer groups using perfectoids; see [16]. It seems to be an open question whether R + is almost Cohen-Macaulay when R has equalcharacteristic zero. If R has mixed characteristic of dimension 3, Heitmann proved that R + is a (p) 1 p ∞ -almost Cohen-Macaulay R-algebra in [28]. Our main concern, inspired also by the recent result of Heitmann and Ma [31], is to extend Heitmann's result to the higher dimensional case, thus giving a positive answer to Roberts' question in mixed characteristic; see Theorem 6.5.
Main Theorem 1. Let (R, m) be a complete Noetherian local domain of mixed characteristic p > 0 with perfect residue field k. Let p, x 2 , . . . , x d be a system of parameters and let R + be the absolute integral closure of R. Then there exists an R-algebra T together with a nonzero element g ∈ R such that the following hold: (1) T admits compatible systems of p-power roots p  ]. In other words, one can find an almost Cohen-Macaulay, normal domain whose p-adic completion is integral perfectoid between R and its absolute integral closure. Using Hochster's partial algebra modification and tilting, one can construct an integral perfectoid big Cohen-Macaulay R-algebra over T ; see [62] for details. In a sense, Main Theorem 1 is regarded as a weak analogue of the mixed characteristic version of a result by Hochster and Huneke. The proof of our result does not seem to come by merely considering decompleted versions of the construction by Heitmann and Ma in [31], due to the difficulty of studying (pg) 1 p ∞ -almost (or (g) 1 p ∞ -almost) mathematics under p-adic completion. For instance, we do not know if it is possible to decomplete André's Riemann's extension theorem (Theorem 8.1), because it is hard to analyze how the functor g − 1 p ∞ ( ) and the p-adic completion are related to each other. 1 This is the main reason one is required to redo the decompletion of André's results in [1] and [2].
Finally, Bhatt recently proved that the absolute integral closure of a complete local domain (R, m) of mixed characteristic has the property that R + /p n R + is a balanced big Cohen-Macaulay R/p n R-algebra for any n > 0; see [10] in which the perfectoidization functor introduced in [12] is used as an essential tool. It will be interesting to know how his methods and results are compared to ours; at present, the authors have no clue. However, it is worth pointing out the following fact.
• The almost Cohen-Macaulay algebra T constructed in Main Theorem 1 is integral over the Noetherian local domain (R, m) and much smaller than the absolute integral closure R + .
In a sense, T is close to being a Noetherian ring. We mention some potential applications of Main Theorem 1.
(1) Connections with the singularities studied in [43] by exploiting the ind-étaleness of R[ 1 pg ] → T [ 1 pg ]. (2) A refined study of the main results on the closure operations in mixed characteristic as developed in [39]. (3) An explicit construction of a big Cohen-Macaulay module from the R-algebra T ; see Corollary 6.10.

Main results on the decompletion of perfectoids and Riemann's extension theorem.
To prove Main Theorem 1, we need to relax the p-adic completeness from Perfectoid Abhyankar's Lemma and incorporate the so-called Witt-perfect condition, which is introduced by Davis and Kedlaya in [18]. Roughly speaking, a Witt-perfect (or p-Witt-perfect) algebra is a p-torsion free ring A whose p-adic completion becomes an integral perfectoid ring. Indeed, Davis and Kedlaya succeeded in proving the almost purity theorem for Witt-perfect rings. The present article is a sequel to authors' previous work [49], in which the authors were able to give a conceptual proof of the almost purity theorem by Davis-Kedlaya by analyzing the integral structure of Tate rings under completion. The advantage of working with Witt-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 is not p-adically complete, but its p-adic completion is integral perfectoid. Let us state the main result; see Theorem 5.20 and Proposition 5.21.
Main Theorem 2. Let A be a p-torsion free algebra over a p-adically separated p-torsion free Witt-perfect valuation domain V of rank 1 admitting a compatible system of p-power roots p 1 p n ∈ V , together with a regular element g ∈ A admitting a compatible system of p-power roots g 1 p n ∈ A. Suppose that the following conditions hold.
(1) A is a p-adically Zariskian and normal ring.
(2) A is a (pg) 1 p ∞ -almost Witt-perfect ring. (3) A is torsion free and integral over a Noetherian normal domain R such that g ∈ R and the height of the ideal (p, g) ⊂ R is 2.
Let us put p n a ∈ A, ∀n > 0 , which is an A-subalgebra of A[ 1 g ]. Let A[ 1 pg ] ֒→ B ′ be a finiteétale extension, and denote by B the integral closure of g In the original version of Perfectoid Abhyankar's Lemma as proved in [1] and [2], it is assumed that A is an integral perfectoid ring, which is necessarily p-adically complete and separated. In Main Theorem 2, this assumption is weakened to p-adic Zariskianness. A detailed study of almost Witt-perfect rings appears in the paper [49]; see also Definition 5.2 below. The functor A → g − 1 p ∞ A is called a functor of almost elements, which is fundamental in almost ring theory. The idea of the proof of Main Theorem 2 is to transport André's original proof to our situation. Here is a summary of ingredients toward the proof: • The almost purity theorem for Witt-perfect rings.
• Descent to Galois extensions of commutative rings.
• Description of the integral structures of affinoid Tate rings via continuous valuations.
• Comparison of integral closure and complete integral closure. The almost purity theorem for Witt-perfect rings is attributed to Davis and Kedlaya; see [18] and [19]. A systematic approach to this important result was carried out in authors' paper [49]. The almost purity theorem yields the assertion of Main Theorem 2 in the case when g = 1. To extend it to the general situations, we need a ring theoretic analogue of Riemann's Extension Theorem. Its perfectoid version has been proved by Scholze in [58], and André used it in the proof of Perfectoid Abhyankar's Lemma in [1]. We establish two types of decompleted variant of it, which are at the core of the technical part of this paper. The first one, which we call Zariskian Riemann's extension theorem, is the following result; see Theorem 5.11. We should remark that it is independent of the theory of perfectoid rings.
Main Theorem 3. Let A be a ring with a regular element t that is t-adically Zariskian and integral over a Noetherian ring. Let g ∈ A be a regular element. Let A j be the Tate ring associated to A[ t j g ], (t) for every integer j > 0 (see Definition 2.8 for Tate rings). Then we have an isomorphism of rings where the transition map A j+1• → A j• is the natural one, and A + For proving Main Theorem 3, a preliminary result Corollary 5.8 is crucial. Recall that an integrally closed domain A is the intersection of all valuation domains that lie between A and the field of fractions; see [65,Proposition 6.8.14] for the proof of this assertion from classical valuation theory. Corollary 5.8 is viewed as a variant of this result for affinoid Tate rings. The assumption that A is t-adically Zariskian and integral over a Noetherian ring is necessary in order to find valuation rings of rank 1 for the proof to work (it is interesting to know to what extent one can relax these assumptions). Main Theorem 3 is also relevant to a standard technique used in nonarchimedean geometry. Indeed, our proof for Proposition 5.7 is inspired by Huber Main Theorem 4. Let A be a p-torsion free algebra over a p-adically separated p-torsion free Witt-perfect valuation domain V of rank 1 admitting a compatible system of p-power roots p 1 p n ∈ V , together with a regular element g ∈ A admitting a compatible system of p-power roots g 1 p n ∈ A. Denote by ( ) the p-adic completion and suppose that the following conditions hold.
p ∞ -almost regular sequence on A (which merely says that g is a (p) 1 p ∞ -almost regular element on A/(p)).
Then the following statements hold. (a) The inclusion map: g ]-algebra isomorphism: Moreover, A j• is Witt-perfect. (c) We have the following identification of rings: (d) There is an injective A-algebra map: The almost isomorphism in the assertion (d) is at the heart of the theorem; notice that in general, inverse limits and taking completion do not commute. Our proof for the assertions (c) and (d) relies on the already-known Riemann's extension theorem for perfectoid algebras. Thus we need to describe the relationship between rational localizations of Tate rings associated to a Witt-perfect ring and the corresponding integral perfectoid ring. The assertions (a) and (b) are consequences of a fine study on it. The (p) 1 p ∞ -almost regularity assumption on the sequence (p, g) ensures that g is (p) 1 p ∞ -almost regular on the p-adic completion A; this is due to Lemma 3.11. Another reason for assuming (p) 1 p ∞ -almost regularity rather than (pg) 1 p ∞ -almost regularity is due to Lemma 5.14. See also Proposition 3.14 as an intermediary step. We remark that there are no common assumptions of both Main Theorem 3 and Main Theorem 4 on the ring A. Let us summarize the content of each section of the present paper.
In §2, we give generalities on almost rings, almost modules and topological rings. We also recall the definitions of perfectoid algebras and their almost analogues whose detailed studies are given in the authors' paper [49].
In §3, some basic results are proved on complete integral closure and its behavior under completion. We stress that the use of "Beauville-Laszlo's lemma" is indispensable for getting meaningful results. This section is intended to give some justification/clarification on the difference between integral and complete integral closures.
In §4, we study some behavior of finiteétale extensions of Tate rings under rational localization. This section is regarded as a complement to [49], and so also includes a brief review of several results in that paper.
In §5, we establish the decompleted variants of Riemann's extension theorem as well as Perfectoid Abhyankar's lemma over almost perfectoid rings. As this section contains quite technical discussions, the reader can skip the details on first reading.
In §6, we give the main applications of the results obtained in the previous sections. The main theorem asserts that one can construct an almost Cohen-Macaulay normal domain between the original complete local domain and its absolute integral closure.
In §7, we prove auxiliary facts on integrality as well as almost integrality via rigid analytic methods, following the book [22].
In §8, we give a complete account of the proof of André's Riemann's extension theorem. To this aim, we also give a proof of the almost vanishing theorem on derived limits, which is discussed in [1]. We hope that this appendix will be helpful for the reader to understand key results in André's original approach.
In §9, we give a brief account on (almost) Galois extensions of commutative rings. These are already treated in André's paper [1] and we omit the proofs.
Caution: In this paper, we take both integral closure and complete integral closure for a given ring extension. This distinction is not essential in our setting in view of Proposition 7.1. However, we opt to formulate the results (mostly) in complete integral closure, because we believe that correct statements of the possible generalizations of our main results without integrality over a Noetherian ring should be given in terms of complete integral closure. The reader is warned that complete integral closure is coined as total integral closure in the lecture notes [8]. We collect notation used in the proof of Theorem 5.20 in Definition 5.12 (see also Remark 5.13).
The almost version of perfectoid or Witt-perfect rings often appears in the following discussions. To the best of authors' knowledge, the first appearance of almost perfectoid rings came from André's work on Perfectoid Abhyankar's Lemma. The reason is that (pg) 1 p ∞ -almost mathematics is essential in André's work, in which case we can only say that the cokernel of the Frobenius map is (pg) 1 p ∞ -surjective. The reader will notice that the base ring in Theorem 5.20 is required to be almost Witt-perfect in order for the proof to work. In a future's occasion, we hope to clarify a real distinction between perfectoid and almost perfectoid rings.

Notation and conventions
We say that a commutative ring A is normal, if the localization A p is an integrally closed domain in its field of fractions for every prime ideal p ⊂ A. For ring maps A → C and B → C, we write A × C B for the fiber product. The completion of a module is always taken to be complete and separated.
2.1. Almost ring theory. We use language of almost ring theory. The most comprehensive references are [23] and [24], where the latter discusses applications of almost ring theory to algebraic geometry and commutative ring theory. Notably, it includes an extension of the Direct Summand Conjecture to the setting of log-regular rings. Throughout this article, for an integral domain A, let Frac(A) denote the field of fractions of A. A basic setup is a pair (A, I), where A is a ring and I is its ideal such that I 2 = I. 2 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 injective (resp. I-almost surjective if the kernel (resp. cokernel) of f is annihilated by I. Moreover, we say that f is an I-almost isomorphism (or simply an almost isomorphism) if both 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 of characteristic 0 with a non-archimedean norm | · | : K → R ≥0 . Fix an element ̟ ∈ K such that |p| ≤ |̟| < 1 and I := n>0 ̟ 1 p n K • (such an element ̟ exists and plays a fundamental role in perfectoid geometry). Set K • := {x ∈ K | |x| ≤ 1} and K •• := {x ∈ K | |x| < 1}. Then K • is a complete valuation domain of rank 1 with field of fractions K and the pair (K • , I) is a basic setup.
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 right adjoint and a left adjoint functors respectively: These are defined by M * := Hom A (I, M 0 ) with M a 0 = M and M ! := I ⊗ A M * . See [23, Proposition 2.2.14 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 with both 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 there is an A-homomorphism M → N that is an I-almost isomorphism. An isomorphism in the category A a − Mod will be denoted by M ≈ N . 3 For technical details, we refer the reader to [23].
Let us recall an explicit description of M * .
Lemma 2.1. Let M be a module over a ring A and let ̟ ∈ A be an element such that A admits a compatible system of p-power roots ̟ 1 p n ∈ A for n ≥ 0. Set I = n>0 ̟ n A with ̟ n := ̟ 1 p n and suppose that ̟ is regular on both A and M . Then the following statements hold: (1) (A, I) is a basic setup. 2 As in [23], we assume that I ⊗A I is flat. For the applications, we only consider the case where I is the filtered colimit of principal ideals; see [23,Proposition 2.1.7]. 3 This symbol is used when there is not necessarily an honest homomorphism between M and N .
(2) There is an equality: Moreover, the natural map M → M * is an I-almost isomorphism. If M is an A-algebra, then M * has an A-algebra structure and the natural map M → M * is an A-algebra map.
Proof. The presentation for M * is found in [57,Lemma 5.3] over a perfectoid field and the proof there works under our setting without any modifications. If M is an A-algebra, then the above presentation will endow M * with an A-algebra structure. In other words, M * is naturally an In the situation of the lemma, we often write M * as n>0 ̟ − 1 p n M or ̟ − 1 p ∞ M to indicate that what basic setup of almost ring theory we are talking. Next we observe that I-almost isomorphy is preserved under pullbacks in the category of (actual) A-algebras.
Lemma 2.2. Let (A, I) be a basic setup. Let f : R → S be an A-algebra homomorphism that admits a commutative diagram of A-algebras: Let ψ : T ′ → T be an A-algebra homomorphism. Then the following assertions hold.
(1) If f is I-almost injective (i.e. Ker(f ) is annihilated by I), then so is the base extension (2) If f is I-almost surjective (i.e. Coker(f ) is annihilated by I), then so is the base extension Proof. We use the explicit description of fiber products: Moreover, since f (r) = 0, xs = 0 for every x ∈ I by assumption. Hence x(t ′ , r) = 0 for every x ∈ I, which yields the assertion.
(2): Pick an element (t ′ , s) ∈ T ′ × T S. Then by assumption, for every x ∈ I, there exists some r x ∈ R such that f (r x ) = xs. Thus we obtain an element (xt ′ , r x ) ∈ T ′ × T R whose image in The following lemma claims that almost isomorphy is preserved under adic completion. Proof. The assertion is equivalent to the assertion that the map ( f ) a : ( M ) a → ( N ) a in A a − Mod is an isomorphism. Since the functor ( ) a commutes with limits, ( f ) a is canonically isomorphic to lim ← −n>0 f a n : lim ← −n>0 (M/J n M ) a → lim ← −n>0 (N/J n N ) a where f n : M/J n M → N/J n N is the A-module map induced by f for every n > 0. Thus it suffices to show that f n is an I-almost isomorphism for every n > 0. It can be easily seen that f n is I-almost surjective because f is so. Let us verify that f n is I-almost injective. First, we have Ker(f n ) = f −1 (J n N )/J n M . Moreover, for an arbitrary ǫ ∈ I, Thus, since f is I-almost injective, for an arbitrary ǫ ′ ∈ I we have Therefore, ǫ ′ ǫ Ker(f n ) = 0. Since I 2 = I, it implies that ǫ ′′ Ker(f n ) = 0 for every ǫ ′′ ∈ I. Hence the assertion follows.

2.2.
Integrality and almost integrality. Here we list several closure operations of rings that will be used frequently.
Definition 2.4. Let R ⊂ S be a ring extension.
(1) An element s ∈ S is integral over R, if ∞ 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. We denote by R + S the integral closure of R in S.
(2) An element s ∈ S is almost integral over R, if ∞ 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. We denote this ring by R * S . If R = T , then R is called completely integrally closed in S. This definition can be extended to any ring map R → S in a natural way, as follows. Let R be a ring, let S be an R-algebra and let s ∈ S be an element. Then we say that s is integral (resp. almost integral) over R, if s is integral (resp. almost integral) over the image of R in S. We should remark that "almost integrality" does not mean "integrality in almost ring theory" in a strict sense, but there is an interesting connection between these two notions; see [49,Lemma 5.3].
From the definition, it is immediate to see that if R is a Noetherian domain and S is the field 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 complete integral closure T of R is not necessarily completely integrally closed in S and such an example was constructed by W. Heinzer [27]. 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 not necessarily mean that b is almost integral over R when b is regarded as an element of S; see [25] for such an example.
We also recall the notion of absolute integral closure due to Artin [5].
Definition 2.5 (Absolute integral closure). Let A be an integral domain. Then the absolute integral closure of A denoted by A + , is defined to be the integral closure of A in a fixed algebraic closure of Frac(A).
2.3. Semivaluation and adic spectra. We need some basic language from Huber's continuous valuations and adic spectra; see [35] and [36]. Continuous valuations are a special class of semivaluations (see Definition 2.6 below) that satisfy a certain topological condition.
The name "semivaluation" refers to the fact that A need not be an integral domain. However, following the usage employed in [35], we will stick to the word "continuous valuation" rather than "continuous semivaluation" for brevity.
In this paper, we mainly consider continuous valuations on Tate rings.
Definition 2.8. Let A be a topological ring.
(1) A is called Tate, if there is an open subring A 0 ⊂ A together with an element t ∈ A 0 such that the topology on A 0 induced from A is t-adic and t becomes a unit in A. A 0 is called a ring of definition and t is called a pseudouniformizer. (2) Let A 0 be a ring and t ∈ A 0 is a regular element. Then the Tate ring associated to (A 0 , (t)) 4 is the ring A := A 0 [ 1 t ] equipped with the linear topology such that {t n A 0 } n≥1 forms a fundamental system of open neighborhoods of 0 ∈ A (it is a unique Tate ring containing A 0 such that A 0 is a ring of definition and t is a pseudouniformizer; see [49,Lemma 2.11]). Let us pick an element | · | ∈ Spa(A, A + ), and set p := {x ∈ A + | |x| = 0}. Then by Lemma 2.9 below, p is a prime ideal, and | · | defines a valuation ring 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 [37]. Proof. This is an easy exercise, using the properties stated in Definition 2.6.
The prime ideal p in Lemma 2.9 is called the support of the semivaluation | · |.

Perfectoid algebras.
Let us recall the notion of perfectoid algebras over a perfectoid field as defined in [57]. These are a special class of Banach algebras; see [49, §2.4] for the definition of Banach rings in this context and how they are related to Tate rings. Definition 2.10 (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.
We will recall the almost variant of perfectoid algebras; see [1]. 4 (t) denotes the principal ideal of A0 generated by t. Notice that the ring A0[ 1 t ] and t-adic topology on A0 are independent of the choice of a generator of the ideal tA0 because t ∈ A0 is regular. 5 This A + should not be confused with the same symbol representing the absolute integral closure in Definition 2.5.
Definition 2.11 (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 following conditions hold: (1) The set of powerbounded elements A • ⊂ A is open and bounded.
(2) The Frobenius endomorphism F rob : Example 2.12. Let A be a perfectoid K-algebra with a nonzero nonunit element t ∈ K • admitting a compatible system of p-power roots {t 1 p n } n>0 . Fix any regular element g ∈ A • that admits a compatible system {g 1 p n } n>0 . Let I := n>0 (tg) 1 p n . Then the pair (A • , I) gives a basic setup, which is a prototypical example that is encountered in this article.

Preliminary lemmas
3.1. Some properties of complete integral closure. Here we investigate several properties of complete integral closure. First, we study how it behaves under separated completion. Thus we start with recalling the following lemma, which is a key for the main results of [7]; see also [64, Tag 0BNR] for a proof and related results.
. Let A be a ring with a regular element t ∈ A and let A be the t-adic completion. Then t is a regular element in A and one has the commutative diagram: Proof. Let (A 0 , (t)) be a pair of definition of A, and A 0 the t-adic completion of A 0 . Then A is the Tate ring associated to ( A 0 , (t)). Hence the first assertion is clear. To check the second assertion, pick a ∈ A such that ψ(a) ∈ A • . Then there exists some l > 0 such that ψ(t l a n ) = t l ψ(a) n ∈ A 0 for every n ≥ 0. Hence t l a n ∈ A 0 for every n ≥ 0 by Lemma 3.1. Therefore, a ∈ A • as desired.
The following lemma is quite useful and often used in basic theory of perfectoid spaces. We take a copy from Bhatt's lecture notes [8].
Lemma 3.3. Let A be a ring with a regular element t ∈ A and let A be the t-adic completion of A. Fix a prime number p > 0. Then the following assertions hold. ( If moreover A admits a compatible system of p-power roots {t If moreover A admits a compatible system of p-power roots {t The following lemma is easy to prove, but plays an important role in our arguments.
Lemma 3.4. Let A be a ring, and let t ∈ A be a regular element. Fix a prime number p > 0. Suppose that A admits a compatible system of p-power roots {t The following corollary is an immediate consequence of Lemma 3.4.
Corollary 3.5. Let A be a ring with a regular element t ∈ A such that A is completely integrally Fix a prime number p > 0. Suppose that A admits a compatible system of p-power roots {t . Now let us discuss complete integral closedness of inverse limits.
Lemma 3.6. 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 integrally Proof. Clearly, lim 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 In the situation of Lemma 2.1, complete integral closedness is preserved under ( ) * .
Lemma 3.7. 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 Therefore, t 1 p k c ∈ B is almost integral over A, as desired.
Next we consider several types of ring extensions. The following lemmas are mainly used in §6.
Lemma 3.8. The following assertions hold.
(1) Let R be a Noetherian integrally closed domain with its absolute integral closure R + and assume that A is a ring such that Then R is also completely integrally closed in S. Proof.
(1): The proof is found in the proof of [60, Theorem 5.9], whose statement is given only for Noetherian normal rings of characteristic p > 0. However, the argument there remains valid for Noetherian normal rings of arbitrary characteristic (see also Proposition 7.1).
(2): For s ∈ S, assume that ∞ 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 our assumption. 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 integral extension. Since Frac(A) is a field, B is a reduced Artinian ring, so that we can write B = Π m i=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 the total ring of fractions of C, which is just B. In other words, C has finitely many minimal prime ideals, because so does B. Then by [64, Tag 030C], C is a finite product of normal domains, which shows that C p is a normal domain for any prime ideal p ⊂ C.  First of all, we record the following fundamental lemma. Lemma 3.11. Let (A, I) be a basic setup and assume that a, b is an I-almost regular sequence on A. Let A denote the a-adic completion of A. Then a and b are I-almost regular elements of A.
Proof. First we prove that a is I-almost regular on A. For any k > 0, the multiplication map A a − → A induces an I-almost injective map A/(a k ) a − → A/(a k+1 ). This forms a commutative diagram: Taking inverse limits along the vertical directions respectively, A a − → A is I-almost injective. Next we prove that b is I-almost regular. Let t ∈ A be such that bt = 0. Then one obviously has bt ∈ a n A for all n > 0. Since b is I-almost regular on A/(a n ) ∼ = A/(a n ), it follows that ǫt ∈ n>0 a n A = 0 for any ǫ ∈ I.
Example 3.12. We give a counterexample to Lemma 3.11 without almost regularity condition. Let us consider the subring: Then it is clear that R is a domain. However, after taking the p-adic completion R, since x ∈ p n R, x becomes zero in R. Therefore, p is a regular element in R, while x is not so.
Let A be a ring with elements f, g ∈ A. Then we can consider the ring of bounded functions defined by a sequence f n , g, denoted by A[ f n g ] as a subring of A[ 1 g ]. In other words, we define where a := m>0 (0 : g m ) as an ideal of A[T ]/(gT − f n ). In the technical part of this paper, the following problem plays a central role.
Problem 1 (Algebraic formulation of Riemann's extension problem). Study the ring-theoretic structure of the intersection In his remarkable paper [58], Scholze studied the perfectoid version of Problem 1, with an application to the construction of Galois representations using torsion classes in the cohomology of certain symmetric spaces. We need the following fact on the description of the ring of bounded functions as a Rees algebra. In the case that the relevant sequence is regular, the proof is found in [64, Tag 0BIQ].
Lemma 3.13. Let (A, I) be a basic setup and let f, g ∈ A be elements such that f, g forms an I-almost regular sequence. Then the ring map for some a ∈ A[T ] and some k ∈ N. Let ǫ ∈ I be any element. Then we want to show that ǫa ∈ (gT − f ). Now we can find b ∈ A[T ] such that g k a = (gT − f )b and write this equation as By the almost regularity of f, g on A[T ], for any ǫ n 1 ∈ I, it follows that . Substituting this back into (3.2), we have ǫ n 1 bf = gf c. As f is I-almost regular, we have ǫ n 1 ǫ n 2 b = ǫ n 2 gc for any ǫ n 2 ∈ I. Since ǫ n 1 ǫ n 2 g k−1 a = ǫ n 1 ǫ n 2 bT − ǫ n 2 f c, we obtain ǫ n 1 ǫ n 2 g k−1 a = ǫ n 2 c(gT − f ). Since ǫ n 1 , ǫ n 2 ∈ I are arbitrary and I = I 2 , we find that ǫg k−1 a ∈ (gT − f ) for any ǫ ∈ I. Arguing inductively on k in view of (3.1), it follows that ǫa ∈ (gT − f ) for any ǫ ∈ I, as desired.
We prove a result which compares the ring of bounded functions under completion. This result will play a crucial role later.
Proposition 3.14. Let the notation and the hypotheses be as in Lemma 3.13. Then for any n ≥ 0, there is an I-almost isomorphism of rings: where ( ) is the f -adic completion.
Proof. In view of Lemma 3.13, we need to show that the natural map is bijective. It suffices to show that for any n > 0, (3.3) is bijective after dividing out by the ideal generated by f n on both sides. So we get which is isomorphic to Since A/(f n ) ∼ = A/(f n ), we are done.

Finiteétale extensions of Tate rings
Here we study some behavior of finiteétale extensions of Tate rings under rational localization.  (1) The topology on B is independent of the choices of A 0 , t and S.
(2) B is a Tate ring with the following property: • for every ring of definition A 0 and every pseudouniformizer t ∈ A 0 of A, there exists a ring of definition B 0 of B that is an integral A 0 -subalgebra of B with finitely many generators and t ∈ B 0 is a pseudouniformizer of B.
Next we recall the definition of (pre)uniformity of Tate rings.
Moreover, we say that A is uniform if A is preuniform and complete and separated.
Permanence of (pre)uniformity is one of the most remarkable features of finiteétale extensions of Tate rings. See [49,Corollary 4.8 (1), (4), and (5)] for the next proposition. (1) B is also preuniform. In particular, for any ring of definition A 0 of A, When one considers separated completion of finiteétale extensions of Tate rings, two types of extension of complete Tate rings appear. The following statement assures that they are isomorphic under the preuniformity assumption. See [49,Corollary 4.10] for the next proposition.
In addition, we record a complement to §3.1. Preuniform Tate rings fit into Galois theory of rings. Proof. Let A 0 be a ring of definition of A and let t ∈ A 0 be a pseudouniformizer of A. As in the proof of [49, Lemma 2.20], we can take a ring of definition B 0 of B that is finitely generated as an A 0 -module and satisfies

4.2.
Rational functions associated to regular sequences. We specialize the above results to study the rings of rational functions associated to regular sequences.
Notation: Let A be Tate ring, (A 0 , (t)) a pair of definition of A, and f, g ∈ A 0 regular elements. Then we define a Tate ring A( f g ) as the Tate ring associated to (A 0 [ f g ], (t)) (see Definition 2.8 (2)), and also define A{ f g } as the separated completion of A( f g ). These Tate rings are independent of the choice of a pair of definition (A 0 , (t)).
One can clarify the relationship between A{ f g } and A{ f g }.
Lemma 4.6. Let A be Tate ring, let (A 0 , (t)) a pair of definition of A, and let f, g ∈ A 0 be regular elements. Let A be the separated completion of A. Suppose that (f, g) forms a regular sequence in is a pair of definition of A{ f g }, the assertion follows.
Let us inspect topological features of finiteétale algebras over the rings of rational functions.
Proposition 4.7. Let A be a Tate ring, and let (A 0 , (t)) be a pair of definition of A. Let g ∈ A 0 be a regular element. Suppose that A( t g ) is preuniform. Let B ′ be a finiteétale A[ 1 g ]-algebra. Let B 0 be the integral closure of A 0 in B ′ , and let B be the Tate ring associated to (B 0 , (t)). Let B ′ t/g be the finiteétale A( t g )-algebra B ′ equipped with the canonical structure as a Tate ring (cf. Lemma 4.1). Then the following assertions hold.
Proof. By Proposition 4.3 (1), ] as rings. By Lemma 4.1 (2), one can take a ring of definition is open in B ′ t/g , as desired. (1) A{ t g } and B{ t g } are the separated completions of A( t g ) and B( t g ), respectively. (2) A{ t g } and B{ t g } are uniform.
g }, and equip B ′ t/g with the canonical structure as a Tate ring that is module-finite over A{ t g } (cf. Lemma 4.1). Then we have isomorphisms of topological rings (where B ′ t/g and B( t g ) denote the separated completions). In particular, B ′ t/g is uniform.
Proof. The assertion (1) follows from Lemma 4.6. Thus, since A( t g ) and B( t g ) are preuniform by Proposition 4.7, the assertion (2) is an isomorphism which restricts to an isomorphism of rings B ′ • are rings of definitions of B ′ t/g and B ′ t/g respectively, (4.2) gives an isomorphism of topological rings The other isomorphism in (4.1) follows from the assertion (1) and Proposition 4.7 (2).
For example, the assumption of regularity in Proposition 4.8 is realized in the following situation. (1) R is a Noetherian N-2 normal domain. 6 (2) There exist t 0 ∈ f −1 (t) and g 0 ∈ f −1 (g) such that the height of the ideal (t 0 , g 0 ) ⊂ R is 2. ( consists of regular elements of A 0 . Then, (t, g) forms a regular sequence on A 0 and B 0 .
To prove Lemma 4.9, we need a more fundamental lemma. Lemma 4.10. Let A be a normal ring that is torsion free and integral over a Noetherian N-2 normal domain R. Assume that (x, y) is an ideal of R such that the height of (x, y) is 2. Then x, y forms a regular sequence on A.
Proof. If A is the zero ring, then any sequence in A forms a regular sequence. Thus we assume that A is not the zero ring below. Notice that then the map R → A is injective because it is torsion free and R is a domain.
Since R is a Noetherian normal domain, every associated prime of the quotient ring R/(x) is minimal by Serre's normality criterion [65,Theorem 4.5.3]. Thus, y ∈ R/(x) is a regular element by the assumption that the height of (x, y) is 2. 7 That is, x, y is a regular sequence on R. Let Frac(R) be the field of fractions of R. Then since R → A is integral and torsion free, it follows that Frac(R) ⊗ R A is the total ring of fractions of A. Let Frac(R) → B be a finite-dimensional subextension of Frac(R) ⊗ R A and denote by R + B be the integral closure of R in B. Since B is a reduced finite Frac(R)-algebra, it is a finite product of fields. Hence, R → R + B is module-finite and R + B is a Noetherian normal ring because R is N-2. We can write In this case, we see that the height of (x, y)R i is 2 for each i and thus, x, y is a regular sequence on R + B . Since A is a normal ring, it can be written as a colimit of such R + B . So we conclude that x, y is regular on A. Now we can deduce Lemma 4.9 easily. 6 A domain R is said to be N-2 if for any finite extension of fields Frac(R) ⊂ L, the ring extension R ⊂ R + L is module-finite. 7 The normality of R is necessary. See [ defines the structure as a torsion free integral R-algebra on B 0 . Therefore, we can apply Lemma 4.10 to deduce the assertion.

A variant of perfectoid Abhyankar's lemma for almost Witt-perfect rings
Let p > 0 be a prime number. For the sake of reader's convenience, we recall the definition of Witt-perfect rings due to Davis and Kedlaya; see [18] and [19].   For applications, we often consider the case that ̟ ∈ A is a regular element and A is (completely) integrally closed in A[ 1 p ]. 8 If one takes ̟ = 1, then it is shown that (̟) 1 p ∞ -almost Witt perfectness coincides with the Witt-perfectness; see [49] for details. Let us recall the following fact; see [49,Proposition 3.20].   (1) In Proposition 5.3, one is allowed to map T to pg ∈ A, in which case A is a (pg) 1 p ∞ -almost Witt-perfect ring for some g ∈ A. We will consider almost Witt-perfect rings of this type. p ∞ ]. Indeed, as A is absolutely integrally closed in its field of fractions, it contains Z + p . Hence A is a Z + p -algebra. 5.1. Variants of Riemann's extension theorems. In the context of commutative ring theory, Riemann's extension theorem often means a kind of theorem that gives a satisfactory answer to Problem 1 in §3.2. Such a theorem for perfectoid algebras is a key result to the proof of the Direct Summand Conjecture and its derived variant; see [2] and [9]. In this subsection, we establish two types of decompleted variant of the perfectoid Riemann's extension theorem; see Theorem 5.11 and Theorem 5.16.

Zariskian Riemann's extension theorem.
We start with recalling the definition of adically Zariskian rings.
Definition 5.5. Let A be ring with an ideal I ⊂ A. Then we say that A is I-adically Zariskian if I is contained in every maximal ideal of A.
Note that Zariskianness is preserved under integral ring maps by [44, §9, Lemma 2]. In other words, for an I-adically Zariskian ring A, any integral A-algebra B is IB-adically Zariskian.
We then introduce an important notion using semivaluations (cf. Definition 2.6). In the following definition, for a semivaluation | · |, we denote by V |·| the valuation ring described in Definition 2.9.
Definition 5.6. Let D ⊂ C be a ring extension and let us set Val(C, D) := | · | | · | is a semivaluation on C such that |D| ≤ 1 and V |·| has dimension ≤ 1 ∼, where ∼ is generated by natural equivalence classes of semivaluations.
Let us prove the following algebraic result.
Proposition 5.7. Let (C, D) be a pair of rings such that C is the localization of D with respect to some multiplicative set consisting of regular elements. Suppose that D is an integral extension of a Noetherian ring R. Fix a (possibly empty) subset S ⊂ D that consists of only regular elements. Then one has D + C = x ∈ C |x| ≤ 1 for any | · | ∈ Val(C, D) such that |g| = 0 for every g ∈ S , where D + C is the integral closure of D in C. In particular, D + C does not depend on the choice of S. Proof. Since the containment ⊂ is clear by the definition of Val(C, D), let us prove the reverse containment ⊃. Let y ∈ C be such that |y| ≤ 1, where | · | ∈ Val(C, D) satisfies |g| = 0 for every g ∈ S. Let D[ 1 y ] be the subring of the localization C[ 1 y ] which is generated by y −1 = 1 y over D. 9 Consider the ring extension D[ 1 y ] ⊂ C[ 1 y ]. First suppose that y −1 is a unit in D[ 1 y ]. Then we can write y = a 0 y n−1 + a 1 y n−2 + · · · + a n−1 for a i ∈ D. Then we have y n − a n−1 y n−1 − · · · − a 0 = 0. Hence y ∈ C is integral over D and y ∈ D + C . To derive a contradiction, suppose that y −1 ∈ D[ 1 y ] is not a unit. We may assume that y is not nilpotent. Choose a prime ideal m ⊂ D[  By our assumption, we have |y| C ≤ 1. Since y −1 ∈ V is in the center, we know |y −1 | C < 1. However, these facts are not compatible with |y| C |y −1 | C = |yy −1 | C = 1 and thus, y −1 ∈ D[ 1 y ] must be a unit, as desired.
The above proposition has the following implication: Keep in mind that A + stands for an open integrally closed subring in a Tate ring A.
Corollary 5.8. Let (A, A + ) be an affinoid Tate ring with a fixed pseudouniformizer t ∈ A + such that A + is t-adically Zariskian and A + is integral over a Noetherian ring. For a regular element g ∈ A + , let us set (C, D) := (A[ 1 g ], A + ). Then we have (5.1) D + C = x ∈ C |x| ≤ 1; ∀ | · | ∈ Val(C, D) such that |t| < 1 .  Proof. Keep the notation as in the proof of Proposition 5.7. The point is that one can choose the valuation domain V so as to satisfy the required property. So assume that y ∈ A[ 1 g ] satisfies |y| ≤ 1 for all | · | ∈ Val(A[ 1 g ], A + ) and y −1 ∈ A + [ 1 y ] is not a unit. Then we can find a maximal ideal m ⊂ A + [ 1 y ] such that y −1 ∈ m, which gives the surjection A + ։ A + [ 1 y ]/m and let n ⊂ A + be its kernel. Then n is a maximal ideal of A + . The element t ∈ A + is in the Jacobson radical by assumption, so we have t ∈ n. There is a chain of prime ideals p ⊂ m ⊂ A + [ 1 y ] such that p is minimal and t, y −1 ∈ m. Then, we have the associated valuation ring (V, | · | V ) and the map A + [ 1 y ]/p ֒→ V . It follows from the above construction that |t| V < 1, establishing (5.1). As t maps into 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 is clear from the construction.  Let s ∈ A 0 be an element such that t ∈ sA 0 . Let X = Spa(A, A + ) and let U be the subspace of X: U := x ∈ X |s| x < 1 for the maximal generization x of x .
Suppose that A 0 is s-adically Zariskian and integral over a Noetherian ring. Then we have where [U ] denotes the maximal separated quotient of U .
Proof. Since we have the containments (the third inclusion holds because | · | x is of rank 1), it suffices to show that By assumption, there exists g ∈ A 0 such that t = sg. Let B be the Tate ring associated to (A 0 , (s)) and Any point | · | ∈ Val(B[ 1 g ], B + ) |s|<1 satisfies that |a| ≤ 1 for any a ∈ A 0 and |t| = |sg| < 1. Thus, since | · | is of rank 1, | · | gives a continuous semivaluation on A such that |A • | ≤ 1. Hence we have a canonical injection . Moreover, B + ⊂ A + , and |s| x = 0 for every x ∈ [U ] because s ∈ A is invertible. Hence (5.4) is also surjective, as desired.
Indeed, the following immediate corollary is already documented in a treatise on rigid geometry.
. Let us discuss an application of the above results. Let A be a ring with regular elements t, g. For every j > 0, we let A j denote the Tate ring associated to (A[ t j g ], (t)). Then the set of A-algebras {A j } j>0 naturally forms an inverse system, where A j+1 → A j is the isomorphism . Then A j+1 → A j is a continuous ring map between Tate rings, so that it induces A j+1• → A j• . There is the following commutative diagram: . Now we can prove the following type of extension theorem, which is fitting into the framework of Zariskian geometry; see [66] for more details.
Theorem 5.11 (Zariskian Riemann's extension theorem). Let A be a ring with a regular element t that is t-adically Zariskian and integral over a Noetherian ring. Let g ∈ A be a regular element. Let A j be the Tate ring associated to A[ t j g ], (t) for every integer j > 0. Then we have an isomorphism of rings Proof. By assumption, we have a canonical ring isomorphism ϕ j : where ϕ is an isomorphism and the vertical maps are injective. Thus it suffices to prove that (5.6) is cartesian. Pick c ∈ A[ 1 tg ] such that ϕ j (c) ∈ A j• for every j > 0. Let us show that c lies in A + by applying Corollary 5.9. For this, we consider the (tg)-adic topology: let A (tg) be the Tate ring associated to A, (tg) (notice that each A j is also the Tate ring associated to A[ t j g ], (tg) ). Let X (tg) = Spa(A (tg) , A + A (tg) ), X j = Spa(A j , A j• ) for each j > 0, and let U be the subspace U = x ∈ X (tg) |t| x < 1 for the maximal generization x of x of X (tg) . Then the underlying ring of A (tg) is equal to A[ 1 tg ], and we have Corollary 5.9. On the other hand, since which factors through [U ] because t ∈ A j is topologically nilpotent. Thus we are reduced to showing that the resulting map f : lim − →j>0 and let | · | x : A (tg) → R ≥0 be a corresponding semivaluation. Let us find some j 0 > 0 such that the composite Since |t| x < 1 and | · | x is of rank 1, there exists some j 0 > 0 such that | t j 0 g | x < 1. Then we have |A[ t j 0 g ]| x ≤ 1 because |A| x ≤ 1 and | · | x is of rank 1. Thus, since any a ∈ A j 0 • is almost integral over A[ t j 0 g ] and | · | x is of rank 1, we have |A j 0 • | x,j 0 ≤ 1. Hence | · | x,j 0 gives the desired point x j 0 ∈ [X j 0 ].

5.1.2.
Witt-perfect Riemann's extension theorem. Next we shall investigate the Riemann's extension problem in the context of Witt-perfect rings by transporting the situation to the case of perfectoid algebras, in which case Riemann's extension theorem has been studied by André, Bhatt and Scholze and known to experts. Let us start setting up some notation.
Notation: Fix a prime number p > 0 and a p-torsion free ring A that admits a compatible system of p-power roots g 1 p n ∈ A for a regular element g ∈ A for n > 0. Moreover, assume the following premises: (1) A is an algebra over a p-adically separated p-torsion free Witt-perfect valuation domain V of rank 1 such that p In this situation, we use the following notation.  (3) For every j > 0, we define B j as the Tate ring associated to (B[ p j g ], (p)) (cf. Definition 2.8 (2)). (4) Let C be a Banach A-algebra, and let || · || C be the norm on it. Put Here we should list several remarks.
Remark 5.13. The notations are as above.
(1) By Proposition 5.3 (and its proof), K is a perfectoid field and A is a (pg) 1 p ∞ -almost perfectoid algebra over K.
(2) It follows from Lemma 3.11 that g ∈ A is a (p) The most important case is when B = A. Let us investigate several properties of A j and A j . First we describe the relationship of them.
Lemma 5.14. In the situation of Definition 5.12, the following assertions hold for every j > 0.
(1) The natural A-algebra map: (2) Let A j be the separated completion of A j . Applying the functor ( ) ⊗ A A[ 1 p ] to (5.7) yields an isomorphism of topological rings: Then we have the following identification of rings: Proof. (1): By assumption, (p, g) is a (p) 1 p ∞ -almost regular sequence on A. Hence by Proposition 3.14, the natural map A p j is a (p) 1 p ∞ -almost isomorphism for every n ≥ 0. Moreover, we have the following commutative diagram: The vertical maps become isomorphisms after applying ( ) ⊗ A A/p m A. Hence they become isomorphisms after p-adic completion. Hence by Lemma 2.3, the lower map (5.7) is also a (p) p ∞ ] via (5.9). Therefore, (5.9) is also a homeomorphism, as desired.
(3): Since A j = A[ 1 pg ] as rings, this assertion immediately follows from the assertion (2) and Corollary 3.2.
Next we discuss preuniformity (resp. uniformity) of A j (resp. A j ).
Proposition 5.15. In the situation of Definition 5.12, the following assertions hold for every j > 0.
(1) A j is a perfectoid K-algebra. In particular, A j is uniform.
p ∞ -almost isomorphism by Lemma 2.3. Hence inverting p yields an isomorphism of topological rings (A ♮ ) j → A j . On the other hand, (A ♮ ) j is a perfectoid K-algebra by [57,Theorem 6.3 (ii)]. Thus, A j is a perfectoid K-algebra.
(2): By the above, (A ♮ ) j → A j restricts to an isomorphism of rings ( by Lemma 2.3, and A • = A. Thus we have the following commutative diagram: where the left vertical map is a (p) 1 p ∞ -almost isomorphism in view of Scholze's result [57,Lemma 6.4]. Hence the right vertical map: p ∞ -almost isomorphism. By considering the composition of (5.7) and (5.12), we obtain the desired (p) as rings (see the proof of Lemma 5.14 (2)). By p ∞ -almost isomorphism (5.13) extends to an isomorphism of rings A p j On the other hand, by Lemma 5.14 (3), A j• = A[ 1 pg ] × A j A j• . By construction, the diagram of rings: is commutative. Thus, the ring map ֒→ A j• . Moreover, by Lemma 2.2, (5.14) is a (p) 1 p ∞ -almost isomorphism. Hence the first assertion follows. In particular, we have

one can apply [49, Proposition 2.4 (1)-(b)] to the inclusion map
On the other hand, Lemma 5.14 (2) allows us to extend (5.7) to a ring isomorphism: By composing the inverse map of (5.15) and (5.16), we obtain the desired isomorphism. In particular, A j• /(p) ∼ = A j• /(p) and A j• /(p 2 ) ∼ = A j• /(p 2 ). Since A j is perfectoid, A j• is Witt-perfect by Proposition 5.3. Hence A j• is also Witt-perfect.
The set of A-algebras {A j } j>0 forms an inverse system, where A j+1 → A j is the natural inclusion defined by (5.17) p j+1 g → p · p j g .
Then A j+1 → A j is a continuous map between Banach K-algebras, so that it induces A j+1• → A j• . Recall that we already defined an inverse system {A j } j>0 in a similar way; see (5.5). After the preparations we have made above, we will establish Witt-perfect Riemann's Extension Theorem (see Theorem 5.16 below). Notice that it is independent of Zariskian Riemann's extension theorem (Theorem 5.11).
Theorem 5.16 (Witt-perfect Riemann's extension theorem). Let A be a p-torsion free algebra over a p-adically separated p-torsion free Witt-perfect valuation domain V of rank 1 admitting a compatible system of p-power roots p 1 p n ∈ V , together with a regular element g ∈ A admitting a compatible system of p-power roots g 1 p n ∈ A. Denote by ( ) the p-adic completion and suppose that the following conditions hold: (1) A is a (pg) (2) (p, g) is a (p) 1 p ∞ -almost regular sequence on A (which merely says that g is a (p) 1 p ∞ -almost regular element on A/(p)).
Then the following assertions hold. (a) We have the following identification of rings: (b) There is an injective A-algebra map: Proof. (a): By Lemma 5.14 (3), we have canonical isomorphisms: On the other hand, it follows from Riemann's extension theorem for (almost) perfectoid K-algebras [1, Théorème 4.2.2] (see Theorem 8.1 for the detailed proof) that there is an A • -algebra isomorphism: Moreover, we have the commutative diagram of rings: where α and β are the unique maps induced by the universal property of localization. Since the composite map g , the assertion follows.
(b): Since the natural maps A → A j• (j > 0) are compatible with the inverse system {A j• } j>0 , one can define a canonical structures as an A-algebra on each one of the rings For a fixed n > 0, consider the exact sequence of inverse systems of A-algebras: Then this induces an injective ring map In view of (5.20), taking the inverse limit with respect to n > 0 yields the composite ring map which gives an injective A-algebra map. Now we want to prove that (5.22) is (g) On the other hand, by the universality of completion again, we have the commutative squares: (j > 0) of which the bottom arrows are compatible with the inverse system { A j• } j>0 . Hence we obtain an A-algebra map which extends to the isomorphism (5.19) as described in the proof of Theorem 8.1. In particular, (5.24) is (g) 1 p ∞ -almost surjective. Here, since lim ← −j>0 A j• is p-adically separated, an extension of the map A → lim ← −j>0 A j• along the completion A → A is unique. Therefore, (5.23) is identified with (5.24). Thus we conclude that (5.23) and hence (5.22) are (g) 1 p ∞ -almost surjective. This completes the proof of the assertion.
By combining Theorem 5.16 with Theorem 5.11, we obtain the following corollary.
Corollary 5.17. Keep the notations and the hypotheses as in Theorem 5. 16. Suppose further that A is p-adically Zariskian and integral over a Noetherian ring. Then we have the equality: where ψ : Proof. By Theorem 5.11 and Theorem 5.16 (a), we have the commutative diagram of rings: where π 1 is the projection map. Since π 1 is injective and Im(π 1 ) = {x ∈ A[ 1 pg ] ψ(x) ∈ g − 1 p ∞ A • }, the assertion follows.
Discussion 5.18. Here is an alternative way to deduce Corollary 5.17. Since A j• is completely integrally closed in A j• [ 1 pg ], it follows that the right-hand side of (5.19) is completely integrally closed after inverting pg by Lemma 3.6. This implies that g − 1 p ∞ A • is completely integrally closed after inverting pg. Thus, A + almost Witt-perfect and completely integrally closed in A. Then is the pair (A, A + ) sheafy, or is it stably uniform? Some relevant results are found in the papers [15] and [45].

5.2.
Witt-perfect Abhyankar's lemma. Now we are prepared to prove the main result, which is a variant of André's Perfectoid Abhyankar's Lemma. Here is the statement of the above main theorem.
Theorem 5.20 (Witt-perfect Abhyankar's lemma). Let A be a p-torsion free algebra over a padically separated p-torsion free Witt-perfect valuation domain V of rank 1 admitting a compatible system of p-power roots p 1 p n ∈ V , together with a regular element g ∈ A admitting a compatible system of p-power roots g 1 p n ∈ A. Suppose that the following conditions hold.
(1) A is a p-adically Zariskian and normal ring.
(3) A is torsion free and integral over a Noetherian normal domain R such that g ∈ R and the height of the ideal (p, g) ⊂ R is 2. We first prove the following preliminary result, which substantially contains the assertion (a) of the theorem.
Proposition 5.21. Keep the notation and the assumption as in Theorem 5.20. Then the following assertions hold.
(1) B is the integral closure of A in B ′ .
(2) For every j > 0, the following assertions hold (see Definition 5.12 for the notation).
(a) Equip B ′ with the canonical structure as a Tate ring that is module-finite over A j (as in Lemma 4.1). Then B ′ = B j as topological rings. In particular, the ring extension A[ 1 pg ] ֒→ B ′ is identified with a continuous ring map A j with the canonical structure as a Tate ring that is module-finite over A j . Then  (2): By Proposition 5.15, A j is preuniform. Hence the assertion (a) follows from Proposition 4.7. In view of Proposition 4.8, to deduce the assertion (b), it suffices to show that (p j , g) forms a regular sequence on A and B. Since A is p-torsion free and p-adically Zariskian, any prime number in A is a regular element. Thus, if the generic characteristic of R is positive, then the R-algebra A and the A-algebra B are the zero rings, where (p j , g) forms a regular sequence. If the generic characteristic of R is 0, then R is N-2 by [21,Theorem 4.6.10], and hence the assertion follows from Lemma 4.9, as desired. The assertion (c) follows from the assertion (b) and [57,Theorem 7.9]. Finally, let us prove the assertion (d Therefore, the first assertion follows. Thus, the second assertion follows from the almost purity theorem for Witt-perfect rings [18,Theorem 5.2] or [19,Theorem 2.9] (see [49,Theorem 5.9] for a conceptual proof).
After applying lim ← − to the standard short exact sequence 0 → B j• /(p p ∞ -almost surjection follows from (5.28): the proof of (1)) and A is (pg) Consider the commutative diagram It suffices to show that in view of (5.30) that By taking the inverse limits over j to the short exact sequence: Let us complete the proof of Theorem 5.20.
Proof of Theorem 5.20. The assertion (a) follows from the assertions (c) and (d) of Proposition 5.21 (4). Let us prove the assertion (b). We fix the notation as in Proposition 5.21. Let us make a reduction by using Galois theory of rings. By decomposing A into the direct product of rings, we may assume and do that A[ 1 pg ] → B ′ is finiteétale of constant rank (indeed, one can check the conditions (1) ∼ (4) remain to hold for each direct factor of the ring A). By Lemma 9.4 applied to the finiteétale extension A[ 1 pg ] ֒→ B ′ = B[ 1 pg ], there is the decomposition  (1)). We will use the consequences of this fact without explicit mention in what follows.
In view of (5.5) and Proposition 5.21 (2)-(a), we obtain the following commutative diagram: Taking inverse limits, we have composite map of rings: where the first isomorphism is due to Theorem 5.11. Similarly, we obtain the compositions of ring maps: By Proposition 5.21 (3), we find that (5.33) and (5.34) are isomorphic to the integral maps A → B and A → C, respectively. Following the convention in Definition 5.12, we set Our goal is the following: p ∞almost finiteétale. As we can treat A → C and B → C in a complete parallel manner in view of Proposition 5.21, we consider only the case A → C in what follows. We use the notation A → C and A → C interchangeably.
Since A j• ∼ = A j• by Proposition 5.15 (4), the map (5.35) is an isomorphism, which induces C j• ∼ = C j• in view of [49,Corollary 4.10]. Thus, G acts on C j• and by applying Lemma 4.5 or Discussion 5.22 (1) below. In particular, A j• → C j• is an integral extension. In summary, ] is a G-Galois covering.
After invoking the notation (5.33) and (5.34), there follows the following (g) 1 p ∞ -almost isomorphisms by applying Theorem 5.16 (b) to A j (resp. C j ): Indeed, this is checked by a chain of almost isomorphisms: Here, the first isomorphism is the p-adic completion of the isomorphism from Theorem 5.11, and the last second isomorphism is due to Riemann's extension theorem [1, Théorème 4.2.2] (see Theorem 8.1 for a self-contained proof). The same reasoning applies to deduce C ≈ C • . In view of (5.37) and applying [1,Proposition 3.3.4], the ring map where the completed tensor product is p-adic. By [1,Proposition 4.4.4], we have C{ p j g } ∼ = C j and C is an A-algebra. Using this, we obtain Putting (5.39) and (5.40) together, we obtain the following (pg) 1 p ∞ -almost isomorphism: p ∞ -almost isomorphism for any m > 0. So this fact combined with the (g) (1) Here is a way to check the isomorphism: ( C j• ) G ∼ = A j• that appears in (5.36). Since inverse limits commutes with taking G-invariants and A j• ∼ = A j• by Proposition 5.15, we have (5.42) ( where ≈ in the middle denotes a (p) 1 p ∞ -almost isomorphism and we reason this as follows: Applying the Galois cohomology H i (G, ) to this exact sequence, we get an injection (C j• ) G /(p m ) ֒→ C j• /(p m ) G whose cokernel embeds into H 1 (G, C j• and p − 1 p ∞ (A j• ) ∼ = A j• by Lemma 3.5. Since the functor p − 1 p ∞ ( ) commutes with taking G-invariants, (5.42) yields an (honest) isomorphism: which proves (5.36). (2) Using (5.36), let us prove that the map p ∞ -almost isomorphism for any integer m > 0. We have already seen the (pg) as wanted.
Problem 2. Does Theorem 5.20 hold true under the more general assumption that A is not necessarily integral over a Noetherian ring?
This problem is related to a possible generalization of Riemann's extension theorem (see Theorem 5.11 and Theorem 5.16) for Witt-perfect rings of general type.
6. Applications of Witt-perfect Abhyankar's lemma 6.1. A construction of almost Cohen-Macaulay algebras. Before proving the main theorem for this section, we recall the definition of big Cohen-Macaulay algebras, due to Hochster.  It is important to keep in mind that the permutation of the sequence x 1 , . . . , x d in the above definition may fail to form an almost regular sequence. We consider the sequence p, x 2 , . . . , x d for the main theorem below.
André's construction: For the applications given below, we take I to be the ideal n>0 ̟ 1 p n T as the basic setup (T, I) for some regular element ̟ ∈ R. Following [2], we introduce some auxiliary algebras. Let W (k) be the ring of Witt vectors for a perfect field k of characteristic p > 0 and let be an unramified complete regular local ring and V j := W (k)[p 1 p j ]. Then V j is a complete discrete valuation ring and set V ∞ := lim − →j V j . Then this is a Witt-perfect valuation domain. For a fixed element 0 = g ∈ A, we set 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 integral closure of A in B jk . Let us also define (6.1) For brevity, let us write Then we have a tower of integral ring maps: Lemma 6.3. Let R be a Noetherian domain with a proper ideal I and let T be a normal ring that is a torsion free integral extension of R. Assume that ̟ ∈ I is a nonzero element such that T admits a compatible system of p-power roots ̟ Proof. In order to prove that T /IT is not (̟) 1 p ∞ -almost zero, it suffices to prove that T m /IT m is not (̟) 1 p ∞ -almost zero, where m is any maximal ideal of T containing IT , since T m /IT m is the localization of T /IT . Then T m is a normal domain that is an integral extension over the Noetherian domain R m∩R , in which I is a proper ideal. To derive a contradiction, we suppose that T m /IT m is (̟ 1 p ∞ )-almost zero. Notice that T m is contained in the absolute integral closure (R m∩R ) + . In particular, it implies that (̟) 1 p n ∈ IT m for all n > 0. Raising p n -th power on both sides, we get by [59,Lemma 4.2]; which is a contradiction.
The big rings A ∞∞ and A ∞g enjoy the following desirable properties. Proposition 6.4. Let the notation be as in (6.1) and (6.2). Then the following assertions hold: (1) A ∞ is completely integrally closed in its field of fractions. It is an integral and faithfully flat extension over A. Moreover, the localization map p ∞ -almost Cohen-Macaulay and Witt-perfect algebra.
(2) A ∞g is a (g) 1 p ∞ -almost Witt-perfect algebra over the Witt-perfect valuation domain V ∞ such that p 1 p n ∈ V ∞ and g 1 p n ∈ A ∞g . Moreover, A ∞g is a (pg) 1 p ∞ -almost Cohen-Macaulay normal ring that is completely integrally closed in A ∞g [ 1 pg ]. In particular, the localization of A ∞g at any maximal ideal is a (pg) integral closure of R. Then there exists an R-algebra T together with a nonzero element g ∈ R such that the following hold: (1) T admits compatible systems of p-power roots p 1 p n , g 1 p n ∈ T for all n > 0.
In the following, we may assume dim R ≥ 2 without loss of generality. By Cohen's structure theorem, there is a module-finite extension x 1 , . . . , x d of R. Then does it hold true that c · (x 1 , . . . , x i ) : R + x i+1 ⊂ (x 1 , . . . , x i )R + for any c ∈ m R + and i = 0, . . . , d − 1?
Bhatt gave an even stronger answer to the above problem in mixed characteristic in [10] by taking x 1 = p n , using prismatic cohomology and mod-p n Riemann-Hilbert correspondence. Namely, p, x 2 , . . . , x d is a regular sequence on R + . In the equal prime characteristic case, Hochster and Huneke already gave a complete answer in [33]. However, almost nothing is known in the equal characteristic zero case. Even in the mixed characteristic case, the above problem is not known to hold true if one starts with an arbitrary system of parameters.
Problem 4. Let T be a big Cohen-Macaulay algebra over a Noetherian local domain (R, m).
• Assume that R has mixed characteristic. Then does T map to an integral perfectoid big Cohen-Macaulay R-algebra? • Assume that R has an arbitrary characteristic. Then does R (or T ) map to a coherent big Cohen-Macaulay R-algebra?
For the coherence of absolutely integrally closed domains, see [51]. Here we mention a few related results.
Proposition 6.7. 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.
(2) B is absolutely integrally closed. In other words, every nonzero monic polynomial in B[X] has a root in B.
In relation to Proposition 6.7 and some observations on p-integral closure as discussed in [17], we prove the following fact, which shows that flatness can be destroyed under taking p-integral closure. We refer the reader to [17, 2.1.7] for details on p-integral closure. Proposition 6.8. Let (R, m) be a non-regular local domain of mixed characteristic p > 0. Then there exists a faithfully flat R-algebra T such that p 1 p n ∈ T for n > 0 and the Frobenius map induces a surjection T /(p 1 p ) → T /(p). Moreover, let T be any R-algebra with the aforementioned properties, and let T be the p-integral closure of T in T [ 1 p ]. Then the p-adic completion of T is integral perfectoid, but T is never flat over R.
Proof. The first assertion is due to Proposition 6.7. It follows from [17, Proposition 2.1.8] that the p-adic completion T is integral perfectoid. Assume that T is flat over R. Since T → T is an integral extension, it follows that T is faithfully flat over R. Moreover, T is faithfully flat over R by [68,Theorem 0.1]. But the main result of [11] forces R to be regular, which contradicts the hypothesis that R is not regular. Theorem 6.9 (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 integrally closed quasi-local domain.
Problem 5. Let (R, m) be a complete Noetherian local domain of mixed characteristic. Then can one construct a big Cohen-Macaulay R-algebra T such that T has bounded p-power roots of p or equivalently, the radical ideal √ pT is finitely generated?
So far, big Cohen-Macaulay algebras constructed using perfectoids necessarily admit p-power roots of p and we do not know if the construction as stated in the problem is possible. Problem 6. Let the notation be the same as that of Theorem 5. 16. Then under what condition is it true that lim ← −j>0 A j• ֒→ lim ← −j>0 A j• is an isomorphism?

Appendix B: Almost vanishing of derived limits
We give a self-contained account of the proof of the Riemann's extension theorem, as proved by André and its consequence on the almost vanishing of the derived limits of a certain tower of perfectoid algebras. This appendix is also meant to help the reader understand André's papers [1] and [2]. See [1, Théorème 4.2.2] and [1, Proposition 4.4.1] for the following results, respectively. Theorem 8.1 (Riemann's extension theorem for perfectoid algebras). Fix a perfectoid K-algebra A, where K is a perfectoid field with a nonzero element ̟ ∈ K • admitting all p-power roots, and let g ∈ A • be an element that admits a compatible system of p-power roots {g 1 p m } m>0 , such that g is a (̟) 1 p ∞ -almost regular element of A • /(̟ r ) for any fixed r ∈ N[ 1 p ]. Then there is an isomorphism: Proof. Throughout the proof, we fix r ∈ N[ 1 p ] and for a given j ∈ N, we set Then the natural ring map η j : A j+1 0 → A j 0 is defined by ̟ j+1 g → ̟ · ̟ j g and hence {A j 0 } j∈N forms an inverse system. 14 Now we claim that the natural map is a (g) 1 Proof. Without loss of generality, we may assume that g is a regular element of A • /(̟ r ) for any r ∈ N. Keep the notation as in Theorem 8.1. Then we have a commutative diagram: Fix an integer m > 0. Let us put N j := Coker(f j ). Then choose c(m) ∈ N according to (8.3) such that the image of the map N j+c(m) → N j is annihilated by g In particular, the right-hand side does not depend on the choice of m ∈ N. Now we claim that To prove this, we may replace the system {N j } j∈N with {N k(m,n) } n∈N to simplify the notation.
Choose any element (β i ) i∈N ∈ j∈N N j and set γ i := g makes sense, where f j,k : N j → N k is the map given above. Then (α k ) k∈N maps to (γ k ) k∈N under the mapping: 1 p ∞ -almost zero, as desired.

Appendix C: Almost Galois extensions
We make use of Galois theory of commutative rings in making reductions in steps of proofs of some results in the present paper. Let A → B be a ring extension and let G be a finite group acting on B as ring automorphisms. Definition 9.1. We say that B is a G-Galois extension of A, if A = B G and the natural ring map Some fundamental results about Galois extensions are found in [1] or [21]. A source of the definition of almost G-Galois extension is [1]. Here we cite some related results for the sake of readers.
Definition 9.2. Let (A, I) be a basic setup and let B → C be an A-algebra map with G acting on C as ring automorphisms. Then we say that B → C is an I-almost G-Galois extension if the natural map B → C induces an I-almost isomorphism B ≈ − → C G and is an I-almost isomorphism. Proposition 9.3. Let (A, I) be a basic setup and let B → C be an A-algebra map with a finite group G acting on C such that B → C factors as B → C G → C. Then the following assertions hold.
(1) If B → C is I-almost G-Galois, then it is I-almost finiteétale of constant rank |G|.
(2) Assume that B → D is an A-algebra map. Let G act on the base change D ⊗ B C through the second factor. If B → C is I-almost G-Galois, then so is D → D ⊗ B C. Conversely, if D is faithfully flat over B and D → D ⊗ B C is I-almost G-Galois, then so is B → C. In the next lemma, S n will denote the group of permutations of n elements.
Lemma 9.4. The following assertions hold.
(1) Assume that B → C isétale of constant rank r. Then there is an S r -Galois extension B → D which factors as B → C → D such that C → D is an S r−1 -Galois extension. (2) Let C be a ring on which a finite group G acts as ring automorphisms. Set B := C G .
Suppose that there is a factorization B → D → C such that G(D) = D and B → D is a G-Galois extension. Then D → C is bijective.