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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.
Research in the Mathematical Sciences – Springer 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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