Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism
Let \(R\) be a ring of positive prime characteristic \(p\). Recall that \(R\) is called weakly \(F\)-regular if all ideals are tightly closed, and \(F\)- regular if, moreover, all localizations of \(R\) are weakly \(F\)-regular. It is an open problem in tight closure theory if weak \(F\)-regularity implies \(F\)-regularity. For a domain \(R\) such that \(R^{1/p}\) is module-finite over \(R\) \((R^{1/p}\) denoting the ring obtained by adjoining to \(R\) all \(p\)-th roots of elements of \(R)\) a third notion has been introduced: \(R\) is said to be strongly \(F\)-regular if for all \(d \in R \backslash \{0\}\) there exists \(q = p^e\) (a power of \(p)\) such that the inclusion \(R \cdot d^{1/q} \subseteq R^{1/q}\) splits as a map of \(R\)-modules. It is known that strong \(F\)-regularity implies \(F\)-regularity, and that a weakly \(F\)-regular Gorenstein ring is strongly \(F\)-regular. The author introduces two concepts, called \(F\)- and \(TF\)-boundedness (``requiring a kind of uniform stability of kernels of certain maps of Koszul cohomology of a family of modules obtained by iterated applications of the Frobenius functor''). These notions turn out to be useful since, in theorem (3.3), it is shown that a weakly \(F\)-regular local domain with some kind of \(TF\)-boundedness is strongly \(F\)-regular, in particular \(F\)-regular. The rest of the paper is devoted to the problem to find classes of rings having this boundedness property. The examples given are, mainly, pure subrings of regular rings (theorem (3.4)), local normal domains of dimension 2 (theorem (4.4)), and certain local Cohen-Macaulay normal domains of dimension 3 (theorem (4.7)). As a consequence one obtains that for a large class of domains of dimension 2 or 3 weak \(F\)-regularity implies \(F\)-regularity.
scientific article; zbMATH DE number 763925