Hilbert-Kunz functions and Frobenius functors

Abstract

We study the asymptotic behavior as a function of e e of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the e e -th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.