Absolute integral closure in positive characteristic

Huneke, Craig; Lyubeznik, Gennady

doi:10.1016/j.aim.2006.07.001

Cited by 46 publications

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“…The argument used in Theorem 5.11 gives a new proof that splinters and derived splinters are the same in characteristic p > 0 for all local rings that are homomorphic image of Gorenstein local rings. First of all it is well known that splinters are Cohen-Macaulay in characteristic p > 0 (we don't need completeness [HL07], [HH92]). Now the only place in the argument that we use the vanishing conditions for maps of Tor and the quasi-Gorenstein hypothesis seriously is in the proof of Claim 5.11.3.…”

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confidence: 99%

“…Now the only place in the argument that we use the vanishing conditions for maps of Tor and the quasi-Gorenstein hypothesis seriously is in the proof of Claim 5.11.3. But this claim is clear in characteristic p > 0 and we give a short argument as follows: by Theorem 2.1 of [HL07] we know that there exists a module-finite extension B of R such that the induced map be the composite map for some splitting g: B ⊗ R T → S. We have the following commutative diagram:…”

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The vanishing conjecture for maps of Tor and derived splinters

2018

J. Eur. Math. Soc.

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We say an excellent local domain (S, n) satisfies the vanishing conditions for maps of Tor, if for every A → R → S with A regular and A → R module-finite torsionfree extension, and every A-module M , the map Tor A i (M, R) → Tor A i (M, S) vanishes for every i ≥ 1. Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions [HH95]. The main theorem of this paper shows that, in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of Bhatt [Bha12]. In particular, rational singularities in characteristic 0 satisfy the vanishing conditions. This greatly generalizes Hochster-Huneke's result [HH95] and Boutot's theorem [Bou87]. Moreover, our result leads to a new (and surprising) characterization of rational singularities in terms of splittings in module-finite extensions.1 It is pointed out in the introduction of [HH93] that by reduction to characteristic p > 0, one can develop the corresponding theory in characteristic 0. The full results in [HH93] are, in some sense, even stronger than Theorem 1.1, but are slightly technical to state here. However we point out that all these (stronger) results can be established by the argument used in [HH95]. Our method can provide generalizations of these results also, both in characteristic p > 0 and characteristic 0, see Remark 5.6.

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The vanishing conjecture for maps of Tor and derived splinters

2018

J. Eur. Math. Soc.

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“…We hasten to remark that there already exist alternative proofs in the literature, all cocycle-theoretic or equational at the core. The approach adopted here follows closely the relatively recent approach from [Huneke and Lyubeznik 2007], the essential new feature being the use of cohomology-annihilation result proven in Theorem 1.1 in place of explicit cocycle manipulations.…”

Section: An Application: Big Cohen-macaulay Algebras In Positive Charmentioning

confidence: 99%

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2012

Algebra Number Theory

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“…algebra in the positive characteristic case. We refer the reader to [19], [22] and [24] for the recent developments on the Cohen-Macaulayness of the absolute integral closure in positive characteristic. The utility of integral perfectoid algebras is contained in the fact that one can reduce the study of big Cohen-Macaulay algebras in mixed characteristic to that of big Cohen-Macaulay algebras in positive characteristic via tilting operations in favorable circumstances (see Corollary 6.7 for more on this).…”

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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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Absolute integral closure in positive characteristic

Cited by 46 publications

References 8 publications

The vanishing conjecture for maps of Tor and derived splinters

The vanishing conjecture for maps of Tor and derived splinters

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

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