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Tight closure and strongly F-regular rings

Tight closure and strongly F-regular rings We describe several aspects of the theory of strongly F-regular rings, including how they should be defined without the hypothesis of F-finiteness, and its relationship to tight closure theory, to F-signature, and to cluster algebras. As a necessary prerequisite, we give a quick introduction to tight closure theory, without proofs, but with discussion of underlying ideas. This treatment includes characterizations, important applications, and material concerning the existence of various kinds of test elements, since test elements play a considerable role in the theory of strongly F-regular rings. We introduce both weakly F-regular and strongly F-regular rings. We give a number of characterizations of strong F-regularity. We discuss techniques for proving strong F-regularity, including Glassbrenner’s criterion and several methods that have been used in the literature. Many open questions are raised. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Tight closure and strongly F-regular rings

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
eISSN
2197-9847
DOI
10.1007/s40687-022-00353-z
Publisher site
See Article on Publisher Site

Abstract

We describe several aspects of the theory of strongly F-regular rings, including how they should be defined without the hypothesis of F-finiteness, and its relationship to tight closure theory, to F-signature, and to cluster algebras. As a necessary prerequisite, we give a quick introduction to tight closure theory, without proofs, but with discussion of underlying ideas. This treatment includes characterizations, important applications, and material concerning the existence of various kinds of test elements, since test elements play a considerable role in the theory of strongly F-regular rings. We introduce both weakly F-regular and strongly F-regular rings. We give a number of characterizations of strong F-regularity. We discuss techniques for proving strong F-regularity, including Glassbrenner’s criterion and several methods that have been used in the literature. Many open questions are raised.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Sep 1, 2022

Keywords: Big test element; Cluster algebra; Completely stable test element; Excellent ring; F-rational ring; F-regular ring; F-pure regular; Frobenius endomorphism; Frobenius functor; F-signature; F-split ring; Glassbrenner criterion; Hilbert–Kunz multiplicity; Peskine–Szpiro functor; Plus closure; Purity; Regular ring; Solid closure; Splinter; Strongly F-regular ring; Tight closure; Very strongly F-regular ring; Weakly F-regular ring; Primary 13A35; Secondary 13D45; 13C14; 13F40; 13H05; 13H10

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