The existence of the \(F\)-signature for rings with large \(\mathbb Q\)-Gorenstein locus
type: Item
instance of: scholarly article
title: The existence of the \(F\)-signature for rings with large \(\mathbb Q\)-Gorenstein locus
zbMATH Open document ID: 1152.13004
author: Ian M. Aberbach
published in: Journal of Algebra
publication date: 2008-05-21T00:00:00Z
review text: Let \((R, m,k)\) be a local ring of positive prime characteristic \(p\) and dimension \(d\). Assume that \(R\) is \(F\)-\textit{finite}, i.e., the Frobenius homomorphism makes \(R\) into a finite \(R\)-algebra. Let \(\alpha=\log_p [k^{1/p}: k]\) and let \(a_q\) denote the number of \(R\)-free direct summands of \(R^{1/q}\) (\(q=p^e\)). The \(F\)-\textit{signature} of \(R\) is \(\lim_{q \to \infty} \frac{a_q}{q^{d+\alpha}}\), provided the limit exists. The author considers the question whether the \(F\)-\textit{signature} exists in the case of strongly \(F\)-regular local rings. Previously established results gave an affirmative answer in the particular cases when \(R\) is Gorenstein, Gorenstein on the punctured spectrum, \(\mathbb{Q}\)-Gorenstein on the punctured spectrum, an affine semigroup ring, and an \(\mathbb{N}\)-graded algebra over a field. All of these are cases when it is know that weak \(F\)-regularity implies strong \(F\)-regularity, and essentially the same techniques can be used to show the existence of the \(F\)-signature. In the paper under review, under a certain boundness condition on local cohomology, the author shows the existence of the \(F\)-\textit{signature} of a strongly \(F\)-regular local ring in the case when the non-\(\mathbb{Q}\)-Gorenstein locus has dimension 1. This is a situation where the boundness assumption is not sufficient to show that weak \(F\)-regularity implies strong \(F\)-regularity.
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5277624
zbMATH Keywords: F-signature
zbMATH Keywords: tight closure
zbMATH Keywords: Hilbert-Kunz multiplicity
zbMATH Keywords: local cohomology
reviewed by: Catalin Ciuperca
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.10.017
OpenAlex ID: W2074231005
cites work: SOME CONDITIONS FOR THE EQUIVALENCE OF WEAK AND STRONG F-REGULARITY
cites work: The structure of \(F\)-pure rings
cites work: When does the F-signature exist?
cites work: The \(F\)-signature and strong \(F\)-regularity.
cites work: Q4274969
cites work: Cyclic Purity Versus Purity in Excellent Noetherian Rings
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: Tight closure and strong F-regularity
cites work: Q4871734
cites work: Two theorems about maximal Cohen-Macaulay modules
cites work: Rational singularities, with applications to algebraic surfaces and unique factorization
cites work: The Hilbert-Kunz function
cites work: Complexes with homology of finite length and Frobenius functors
cites work: The F-signature of an affine semigroup ring
cites work: F-rational rings have rational singularities
cites work: Simplicity of Rings of Differential Operators in Prime Characteristic
cites work: Local cohomology of Frobenius images over graded affine algebras
cites work: Minimal relative Hilbert-Kunz multiplicity
cites work: Observations on the \(F\)-signature of local rings of characteristic \(p\)
DOI: 10.1016/J.JALGEBRA.2007.10.017
Recommended article: \(F\)-signature of graded Gorenstein rings
Recommended article: Observations on the \(F\)-signature of local rings of characteristic \(p\)
Recommended article: The signature of the Chern coefficients of local rings
Recommended article: A Characterization of Gorenstein Rings and Grothendieck's Conjecture
Recommended article: -SIGNATURE UNDER BIRATIONAL MORPHISMS
Recommended article: Weak F-regularity deforms in \(\mathbb{Q}\)-Gorenstein rings
Recommended article: Generalized \(F\)-signatures of Hibi rings
Recommended article: \(f\)-rings and the Stone-Weierstrass theorem
Recommended article: The F-rational signature and drops in the Hilbert-Kunz multiplicity
Recommended article: The F-signature of an affine semigroup ring
description: scientific article; zbMATH DE number 5277624