Closure operations in complete local rings of mixed characteristic

Jiang, Zhan

doi:10.1016/j.jalgebra.2021.03.038

Cited by 3 publications

(5 citation statements)

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“…We review some preliminaries in §2. We show that the definition of symbolic powers in (1.1) matches the definition used in [Mur] (Lemma 2.1) and define full extended plus (epf) closure [Hei01] and weak epf (wepf) closure [Jia21]. We state Dietz's axioms for closure operations [Die10] and R.G.…”

Section: Introductionmentioning

confidence: 83%

“…Our strategy in this paper utilizes epf closure in conjunction with Jiang's weak epf (wepf) closure [Jia21] and R.G. 's results on closure operations that induce big Cohen-Macaulay algebras [RG18].…”

Section: Introductionmentioning

confidence: 99%

“…'s results to work around the fact that epf closure does not localize [Hei01, p. 817], and hence Hochster and Huneke's unpublished proof of Theorem A using plus closure in equal characteristic p > 0 [HH02, p. 353] cannot readily be adapted to the mixed characteristic case. Following [Jia21], for a domain R as in the previous paragraph, the weak epf (wepf) closure of an ideal I ⊆ R is…”

Section: Introductionmentioning

confidence: 99%

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Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings

Murayama¹

2022

Preprint

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We give short, closure-theoretic proofs for uniform bounds on the growth of symbolic powers of ideals in regular rings. The author recently proved these bounds in mixed characteristic using a new version of perfectoid/big Cohen-Macaulay test ideals, with special cases obtained earlier by Ma and Schwede. In mixed characteristic, we instead use Heitmann's full extended plus (epf) closure, Jiang's weak epf (wepf) closure, and R.G.'s results on closure operations that induce big Cohen-Macaulay algebras. Our strategy also applies to any Dietz closure satisfying R.G.'s algebra axiom and a Briançon-Skoda-type theorem, and hence yields new proofs of these results on uniform bounds on the growth of symbolic powers of ideals in regular rings of all characteristics. In equal characteristic, these results on symbolic powers are due to Ein-Lazarsfeld-Smith, Hochster-Huneke, Takagi-Yoshida, and Johnson.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings

Murayama¹

2022

Preprint

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“…We mention some potential applications of Main Theorem 1. Connections with the singularities studied in [39] by exploiting the ind-étaleness of . A refined study of the main results on the closure operations in mixed characteristic as developed in [35]. An explicit construction of a big Cohen–Macaulay module from the R -algebra T (see Corollary 6.10).…”

Section: Introductionmentioning

confidence: 99%

“…A refined study of the main results on the closure operations in mixed characteristic as developed in [35]. 3.…”

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confidence: 99%

A Variant of Perfectoid Abhyankar’s Lemma and Almost Cohen–macaulay Algebras

NAKAZATO,

SHIMOMOTO

2023

Nagoya Math. J.

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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.

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A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras

Nakazato¹,

Shimomoto²

2020

Preprint

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In this paper, 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 algebra such that the Frobenius map is surjective modulo p. This is seen as a mixed characteristic analogue of the classical result of Hochster-Huneke for the absolute integral closure in characteristic p > 0. To achieve this goal, we establish the Witt-perfect version of André's perfectoid Abhyankar's lemma and Riemann's extension theorem. The advantage of Witt-perfet rings is that they are not assumed to be p-adically complete.

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Closure operations in complete local rings of mixed characteristic

Cited by 3 publications

References 10 publications

Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings

Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings

A Variant of Perfectoid Abhyankar’s Lemma and Almost Cohen–macaulay Algebras

A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras

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