On the existence of 𝐹-thresholds and related limits

Stefani, Alessandro De; Núñez-Betancourt, Luis; Pérez, Felipe

doi:10.1090/tran/7176

Cited by 20 publications

(14 citation statements)

References 58 publications

(65 reference statements)

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“…[HMTW08, DSNBP16] Suppose that is a standard graded -algebra. If , the -threshold of with respect to is defined by When , we call the diagonal -threshold of , and we denote it by .…”

Section: Properties Of -Thresholdsmentioning

confidence: 99%

F-THRESHOLDS OF GRADED RINGS

Stefani¹,

Núñez-Betancourt²

2016

Nagoya Math. J.

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Abstract. The a-invariant, the F -pure threshold, and the diagonal F -threshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F -regular rings. In this article, we prove that these relations hold only assuming that the algebra is F -pure. In addition, we present an interpretation of the a-invariant for F -pure Gorenstein graded K-algebras in terms of regular sequences that preserve F -purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo-Mumford regularity, and Serre's condition S k . We also present analogous results and questions in characteristic zero.

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“…[HMTW08, DSNBP16] Suppose that is a standard graded -algebra. If , the -threshold of with respect to is defined by When , we call the diagonal -threshold of , and we denote it by .…”

Section: Properties Of -Thresholdsmentioning

confidence: 99%

F-THRESHOLDS OF GRADED RINGS

Stefani¹,

Núñez-Betancourt²

2016

Nagoya Math. J.

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“…The limit of {ν I m (q)/q} as q → ∞ is called the diagonal F -threshold of R with respect to I , denoted c I (R) . The existence of such a limit was recently proved by De Stefani, Núñez-Betancourt and Pérez [18] (They proved a much more general result that F -threshold always exists). We also refer to [3,4,12] for some interesting computations especially related to diagonal F -thresholds.…”

Section: Lemma 33 Given the Following Commutative Diagram (With Exacmentioning

confidence: 92%

“…For another application of Theorem 3.2, we need to recall the definition of diagonal F -threshold in [13,Definition 2.2], which is a special case of the concept of F -threshold in a more general setting, see [2,6,7,[16][17][18].…”

Section: Lemma 33 Given the Following Commutative Diagram (With Exacmentioning

confidence: 99%

Asymptotic socle behaviors for cones over curves in positive characteristic

Li¹

2020

Turk J Math

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“…. , x d ) is defined as the limit (the fact that the limit exists was settled fairly recently in [5] We will show that a special case of the conjecture follows from Inequality A. The proof is adapted from [11,Theorem 3.3].…”

Section: A Conjectured Inequality On F -Thresholdsmentioning

confidence: 95%

The multiplicity and the number of generators of an integrally closed ideal

Dao¹,

Smirnov²

2019

jsing

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Let (R, m) be a Noetherian local ring and I a m-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if Spec R has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. In addition, we can classify when the inequality always hold for a Cohen-Macaulay R of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on pg ideals by Okuma-Watanabe-Yoshida, a conjecture of Huneke, Mustaţǎ, Takagi, and Watanabe on F -threshold, multiplicity of the fiber cone, and the h-vector of the associated graded ring. statement in the dimension two case of Theorem 1.1. We are also grateful to

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On the existence of 𝐹-thresholds and related limits

Cited by 20 publications

References 58 publications

F-THRESHOLDS OF GRADED RINGS

F-THRESHOLDS OF GRADED RINGS

Asymptotic socle behaviors for cones over curves in positive characteristic

The multiplicity and the number of generators of an integrally closed ideal

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