Le lemme d’Abhyankar perfectoide

André, Yves

doi:10.1007/s10240-017-0096-x

Cited by 33 publications

(76 citation statements)

References 22 publications

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“…As observed by Shimomoto [38,Proposition 4.9], the ring A is perfectoid and the map R → A is faithfully flat (see also [4,Example 3.4.5], [9,Proposition 5.2]). In terms of…”

Section: Recollections On Perfect and Perfectoid Ringsmentioning

confidence: 85%

Regular rings and perfect(oid) algebras

Bhatt

Iyengar

2019

Communications in Algebra

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We prove a p-adic analog of Kunz's theorem: a p-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness of projective dimension of finitely generated modules over noetherian rings via maps to perfectoid rings. We also establish a version of the p-adic Kunz's theorem where the flatness hypothesis is relaxed to almost flatness.Date: September 11, 2018. 2010 Mathematics Subject Classification. 13D05 (primary); 13A35, 13D22, 13H05 (secondary).

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“…As observed by Shimomoto [38,Proposition 4.9], the ring A is perfectoid and the map R → A is faithfully flat (see also [4,Example 3.4.5], [9,Proposition 5.2]). In terms of…”

Section: Recollections On Perfect and Perfectoid Ringsmentioning

confidence: 85%

Regular rings and perfect(oid) algebras

Bhatt

Iyengar

2019

Communications in Algebra

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“…More on the history of this conjecture and its centrality amongst the 'homological conjectures' in commutative algebra can be found in [Ho3]. The result above is proven by André [An2] using [An1].…”

Section: Introductionmentioning

confidence: 87%

On the direct summand conjecture and its derived variant

Bhatt

2017

Invent. math.

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André recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where "discriminants" acquire all p-power roots.In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze's Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.

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“…Noetherian local normal domain of mixed characteristic with perfect residue field without loss of generality. Let S := R be as in Theorem 6.1 (1). Then we get an R ♭ -algebra B(R ♭ )…”

Section: Main Theoremsmentioning

confidence: 99%

“…The proof of Main Theorem 1 relies heavily on the following remarkable result of André, together with the construction of a certain seed algebra over the Fontaine ring, using the Frobenius map. Thus, Main Theorem 1 is regarded as a refinement of Theorem 1.2, which is found in [2] and its proof uses a deep theorem, Perfectoid Abhyankar's Lemma as proved in [1]. The existence of big Cohen-Macaulay algebras can be deduced from Theorem 1.2 by combining Hochster's method of (partial) algebra modifications.…”

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confidence: 98%

Integral perfectoid big Cohen–Macaulay algebras via André’s theorem

Shimomoto

2018

Math. Ann.

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The main result of this article is to prove that any Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay algebra. The proof of this result is based on the construction of almost Cohen-Macaulay algebras in mixed characteristic due to Yves André. Moreover, we prove that the absolute integral closure of a complete Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay algebra.Macaulay R-module with respect to x. The notion of big Cohen-Macaulay modules was introduced by Hochster in 1970's for the purpose of studying the homological conjectures (see [16] for a brief survey and [18] in the equal characteristic case). Hochster proved that any local ring (R, m) of equal characteristic admits a big Cohen-Macaulay module. The issue whether such a module exists in mixed characteristic has remained unclear for a long time. Quite recently, André proved that big Cohen-Macaulay algebras exist in the mixed 1 2 K.SHIMOMOTO characteristic case in [1] and [2], which also implies that the Direct Summand Conjecture is fully settled (see [7] for the derived variant of this conjecture).Theorem 1.1 (Y. André). Every complete Noetherian local domain of mixed characteristic admits a big Cohen-Macaulay algebra.This theorem can be deduced from Theorem 1.2, which we explain below. The primary aim of this article is to show abundance of big Cohen-Macaulay algebras with certain distinguished properties based on Theorem 1.2 stated below. First, we state the main theorem (see Theorem 6.3).Main Theorem 1. Let (R, m) be a Noetherian local domain of mixed characteristic. Then there exists an R-algebra T such that T is an integral perfectoid big Cohen-Macaulay R-algebra. Moreover, we have the following assertions:(1) Assume that R is a complete Noetherian local domain of mixed characteristic with perfect residue field. Then there exists an integral perfectoid big Cohen-Macaulay R-algebra T with the property that R → T factors as R → S → T , such that S is an integral pre-perfectoid normal domain that is integral over R and(2) Assume that R is a complete Noetherian local domain of mixed characteristic. Let B be an integral almost perfectoid, almost Cohen-Macaulay R-algebra such that R → B factors as R → S → B and S is an integral perfectoid algebra containing compatible systems of elements: {p 1 p n } n≥0 , {x 1 p n 2 } n≥0 , . . . , {x 1 p nd } n≥0 for a system of parameters p, x 2 , . . . , x d of R. Then there is a ring homomorphism S → T such that T is an integral perfectoid big Cohen-Macaulay R-algebra.One can indeed prove the following stronger result (see Corollary 6.5 and Remark 6.6).Main Theorem 2. Let (R, m) be a Noetherian local domain of mixed characteristic and let R + be its absolute integral closure. Then R + maps to an integral perfectoid big Cohen-Macaulay R-algebra.For basic notion and notations, we refer the reader to § 2. An integral perfectoid algebra is a p-torsion free, p-adically complete algebras on which the Frobenius endomorphism is su...

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Le lemme d’Abhyankar perfectoide

Cited by 33 publications

References 22 publications

Regular rings and perfect(oid) algebras

Regular rings and perfect(oid) algebras

On the direct summand conjecture and its derived variant

Integral perfectoid big Cohen–Macaulay algebras via André’s theorem

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