Tight closure and strongly F-regular rings
Tight closure and strongly F-regular rings
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DOI：
10.1007/s40687-022-00353-z
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发表时间：
2022-08
影响因子：
1.2
通讯作者：
M. Hochster
中科院分区：
文献类型：
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作者：
M. Hochster
我们描述了强F-正则环理论的几个方面，包括在没有F-有限性假设的情况下如何定义它们，以及它们与紧闭包理论、F-签名和簇代数的关系。作为一个必要的先决条件，我们给出了一个快速介绍紧闭包理论，没有证明，但与讨论的基本思想。这种治疗包括特征，重要的应用，和材料有关的存在各种测试元素，因为测试元素发挥了相当大的作用，在理论的强F-正则环。我们引入了弱F-正则环和强F-正则环。给出了强F-正则性的若干刻画。我们讨论了证明强F-正则性的技术，包括Glassbrenner准则和文献中使用的几种方法。提出了许多悬而未决的问题。
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.