Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

Polstra, Thomas; Quy, Pham Hung

doi:10.1016/j.jalgebra.2019.03.015

Cited by 20 publications

(11 citation statements)

References 30 publications

(14 reference statements)

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“…Recently, many applications of this fact have been found [4,12,13]. It should be noted that Nagel-Schenzel's theorem was proved by using spectral sequences.…”

Section: Nagel-schenzel's Isomorphismmentioning

confidence: 97%

Notes on the Frobenius test exponents

Huong

Quy

2019

Communications in Algebra

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“…Recently, many applications of this fact have been found [4,12,13]. It should be noted that Nagel-Schenzel's theorem was proved by using spectral sequences.…”

Section: Nagel-schenzel's Isomorphismmentioning

confidence: 97%

Notes on the Frobenius test exponents

Huong

Quy

2019

Communications in Algebra

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“…Our full result, Theorem 4.3, answers positively a question of the second author [30, Question 1] and it immediately implies one implication ((3) ⇒ (2)) in the Main Theorem. Our main technique is the relative Frobenius action on local cohomology introduced in [28], which turns out to be very useful in the study of tight closure of parameter ideals.…”

Section: Relative Frobenius Action On Local Cohomologymentioning

confidence: 99%

A Buchsbaum theory for tight closure

Ma¹,

Quy²

2021

Preprint

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A Noetherian local ring (R, m) is called Buchsbaum if the difference e(q, R) − ℓ(R/ q), where q is an ideal generated by a system of parameters, is a constant independent of q. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring (R, m) of prime characteristic p > 0 and dimension at least one, the difference e(q, R) − ℓ(R/ q * ) is independent of q if and only if the parameter test ideal τ par (R) contains m. We also provide a characterization of this condition via derived category which is analogous to Schenzel's criterion for Buchsbaum rings.2010 Mathematics Subject Classification. Primary 13A35; Secondary 13H10. Key words and phrases. tight closure, system of parameters, parameter test ideal, Buchsbaum ring, generalized Cohen-Macaulay ring.LM is partially supported by NSF Grant DMS #1901672, NSF FRG Grant #1952366, and a fellowship from the Sloan Foundation. PHQ is partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2020.10.1 This characterization of Buchsbaum ring is well-known to experts but we cannot find an explicit reference in the literature, thus we include a short explanation in Remark 3.4.

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“…In this subsection, we recall the notion of relative Frobenius action on local cohomology which was introduced in [14] by Polstra and Quy in study F -nilpotent rings. Let K ⊆ I be ideals of R. The Frobenius endomorphism F : R/K → R/K can be factored as composition of two natural maps:…”

Section: The Relative Frobenius Action On Local Cohomologymentioning

confidence: 99%

Frobenius test exponent for ideals generated by filter regular sequences

Huong¹,

Quy²

2021

Preprint

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Let (R, m) be a Noetherian local ring of prime characteristic p > 0, and t an integer such that H j m (R)/0 F H j m (R) has finite length for all j < t. The aim of this paper is to show that there exists an uniform bound for Frobenius test exponents of ideals generated by filter regular sequences of length at most t. H i m (R) , consists nilpotent elements of H i m (R) under the Frobenius action. The Hartshorne-Speiser-Lyubeznik number

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Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

Cited by 20 publications

References 30 publications

Notes on the Frobenius test exponents

Notes on the Frobenius test exponents

A Buchsbaum theory for tight closure

Frobenius test exponent for ideals generated by filter regular sequences

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