Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism.

Abstract

In this paper all rings are commutative with identity and Noetherian of positive prime characteristic. We introduce the notions of F- and TF-boundedness. Both definitions require a kind of uniform stability of kernels of certain maps of Koszul cohomology of a family of modules obtained by iterated application of the Frobenius functor. From the definitions it is clear that regular rings are both F- and TF-bounded. Consequently we are able to see that pure subrings of regular rings are also both F- and TF-bounded. Some other special cases in which we are able to prove F- and TF-boundedness are also given. In addition, we show that under certain conditions, weak F-regularity coupled with TF-boundedness implies strong F-regularity. Finally, we see that some local Cohen-Macaulay normal domains of Krull dimension two or three have a system of parameters which give TF-boundedness on certain ideals. Consequently we obtain the result that if a domain has a canonical module, is weakly F-regular, and is such that the Frobenius endomorphism is finite, and if the domain has dimension two or if it has dimension three and has isolated non-Gorenstein points, then it is strongly F-regular, and hence F-regular.