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Tight closure and strongly F-regular rings

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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.

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Notes

  1. An affine algebra over a field of characteristic 0 is said to have rational singularities if it is normal and the higher direct images of the structure sheaf of a desingularization are 0. This implies the Cohen–Macaulay property.

  2. More generally, if there are only isolated non-\(\mathbb {Q}\)-Gorenstein point [70, 91].

  3. Since R is F-finite it is excellent, and one may even choose \(c \in R^{\circ }\) such that \(R_c\) is regular. Thus, such choices of c always exist.

  4. Related results are given in [87].

  5. For the somewhat technical explanation of what it means for a representation to be good, we refer the reader to [29].

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  1. Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI, 48109–1043, USA

    Melvin Hochster

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This paper is dedicated to Jürgen Herzog on the occasion of his 80th birthday, in celebration of his fundamental contributions to commutative algebra.

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Hochster, M. Tight closure and strongly F-regular rings. Res Math Sci 9, 56 (2022). https://doi.org/10.1007/s40687-022-00353-z

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