Galois extensions, plus closure, and maps on local cohomology

Sannai, Akiyoshi; Singh, Anurag K.

doi:10.1016/j.aim.2011.12.021

Cited by 13 publications

(11 citation statements)

References 17 publications

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“…One may wonder whether Proposition 4.2 can be refined to show the existence of generically separable finite surjective maps that kill the relevant cohomology groups. In the local algebra setting, one can indeed do so by [SS12,Theorem 1.3]. Globally, however, requiring separability is too strong.…”

Section: B Bhattmentioning

confidence: 99%

Derived splinters in positive characteristic

Bhatt

2012

Compositio Math.

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This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e. schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning the extension of Serre/Kodaira-type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove 'up to finite cover' analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjecture.

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Section: B Bhattmentioning

confidence: 99%

Derived splinters in positive characteristic

Bhatt

2012

Compositio Math.

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“…In fact, we require a variant with an even stronger conclusion; namely, that the guaranteed finite extension may be assumed separable. This generalization follows from a recent result of A. Sannai and A. Singh [SS12]. ) is zero.…”

Section: F -Rationality Via Alterationsmentioning

confidence: 60%

“…We do not have a good answer to this question. The key point in our construction is repeated use of the Equational Lemma [HH92, HL07,SS12]. The procedure in that Lemma is constructive.…”

Section: Further Questionsmentioning

confidence: 99%

“…The central ingredients in its proof are the argument of K. Smith [Smi97b] that F -rational singularities are pseudo-rational, and the work of C. Huneke and G. Lyubeznik on annihilating local cohomology using finite covers [HL07] (cf. [HH92,HY11,SS12]). The proof of the Main Theorem additionally utilizes transformation rules for test ideals under finite morphisms [ST10].…”

Section: Introductionmentioning

confidence: 99%

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F -singularities via alterations

Blickle¹,

Schwede²,

Tucker³

2015

ajm

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Abstract. We give characterizations of test ideals and F -rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly, we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further demonstrate how these vanishing theorems may be used to extend sections.

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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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confidence: 99%

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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Galois extensions, plus closure, and maps on local cohomology

Cited by 13 publications

References 17 publications

Derived splinters in positive characteristic

Derived splinters in positive characteristic

F -singularities via alterations

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

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