Rings of Frobenius operators
type: Item
instance of: scholarly article
title: Rings of Frobenius operators
zbMATH Open document ID: 1332.13005
DOI: 10.1017/S0305004114000176
author: Mordechai Katzman
author: Karl Schwede
author: Anurag K. Singh
author: Wenliang Zhang
published in: Mathematical Proceedings of the Cambridge Philosophical Society
publication date: 2014-08-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.6147
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 6333636
zbMATH Keywords: injective hull
zbMATH Keywords: Frobenius operators
zbMATH Keywords: twisted construction
zbMATH Keywords: \(\mathbb{Q}\)-Gorenstein
zbMATH Keywords: F-jumping numbers
zbMATH Keywords: gauge bounded
MaRDI profile type: Publication
arXiv ID: 1304.6147
cites work: Q5431016
cites work: On graded rings. I
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cites work: Test ideals via algebras of 𝑝^{-𝑒}-linear maps
cites work: A non-finitely generated algebra of Frobenius maps
cites work: An elementary approach to \(L\)-functions \(\text{mod }p\)
cites work: Simplicity of Rings of Differential Operators in Prime Characteristic
cites work: Frobenius and Cartier algebras of Stanley-Reisner rings
cites work: Test ideals in non-$\mathbb{Q}$-Gorenstein rings
cites work: On the commutation of the test ideal with localization and completion
cites work: \(D\)-modules over rings with finite F-representation type
review text: Let \(R\) be a local ring of prime characteristic and \(E\) be the injective hull of its residue field. Let \(\mathcal F (E)\) be the ring of Frobenius operators of \(E\) over \(R\). This paper studies the finite generation of \(\mathcal F (E)\) over \(\mathcal F^0(E) = \text{Hom}_R(E,E)\) as a ring extension.NEWLINENEWLINEThe authors introduce a twisted construction \(\mathcal T (\mathcal R)\) associated to an \(\mathbb{N}\)-graded commutative ring \(\mathcal R\). They prove that, when \(R\) is normal and complete, the ring of Frobenius operators is isomorphic to the twisted construction associated to the anticanonical cover of \(R\). Using this result, they give an alternate proof to a result by \textit{K. Schwede} [Trans. Am. Math. Soc. 363, No. 11, 5925--5941 (2011; Zbl 1276.13008)] that states that, for a normal \(\mathbb{Q}\)-Gorenstein ring \(R\), if the order of the canonical module in the divisor class group is relatively prime to \(p\), then \(\mathcal F(E)\) is a finitely generated ring extension of \(\mathcal F^0(E)\). A conjecture is stated that asserts the converse.NEWLINENEWLINEThe paper also proves a conjecture of \textit{M. Katzman} [Proc. Am. Math. Soc. 138, No. 7, 2381--2383 (2010; Zbl 1192.13004)] confirming that determinantal ring obtained by moding out size \(2\) minors of a \(2 \times 3\) matrix of indeterminates has an infinitely generated ring of Frobenius operators over \(\mathcal F^0(E)\). The last two sections contain applications to the \(F\)-jumping numbers of ideals in a ring. Let \(R\) be a normal \(\mathbb{N}\)-graded that is finitely generated over an \(F\)-finite ring \(R_0\). Assume that the anticanonical cover of \(R\) is finitely generated as an \(R\)-algebra. It is shown in the paper that the set of \(F\)-jumping numbers of ideals in \(R\) is a subset of the real numbers with no limit points. In the last section, the authors consider a standard graded polynomial ring \(A\) over an \(F\)-finite field of prime characteristic and …
reviewed by: Florian Enescu
description: scientific article; zbMATH DE number 6333636