Extension of weakly and strongly F-regular rings by flat maps
This paper is about the theory of tight closure in positive prime characteristic \(p\), and has ramifications for the existence of test elements. The main results are about the passage of weak/strong F-regularity under a flat map. By Singh's counterexample [\textit{A. K. Singh}, Am. J. Math. 121, No. 4, 919-929 (1999; Zbl 0946.13002)] it is known that flatness is an insufficient assumption. The author's paper is a considerable generalization of the previously known cases of when F-regularity lifts. In particular, he proves the following: NEWLINENEWLINENEWLINELet \((R,m) \to (S,n)\) be a flat local map of Noetherian rings of characteristic \(p\) and with Gorenstein closed fiber. If \(R\) is weakly F-regular and Cohen-Macaulay, then \(S\) is weakly F-regular if one of the following conditions is satisfied: NEWLINENEWLINENEWLINE1. \(R\) and \(S\) have a common test element and \(S/mS\) is F-injective; NEWLINENEWLINENEWLINE2. \(S/mS\) is F-rational and \(S \setminus mS\) contains a test element for \(S\); NEWLINENEWLINENEWLINE3. \(R\) is excellent and \(S/mS\) is F-rational.
scientific article; zbMATH DE number 1657187