Coherent functors and asymptotic stability

Abstract

Asymptotic properties of high powers of an ideal related to a coherent functor F are investigated. It is shown that when N is an artinian module the sets of attached prime ideals AttAF(0:Nan) are the same for n large enough. Also it is shown that for an artinian module N if the modules F(0:Nan) have finite length and for a finitely generated module M if the modules F(M/anM) have finite length, their lengths are given by polynomials in n, for large n. When A is local it is shown that, the Betti numbers βi(F(M/anM)) and the Bass numbers μi(F(M/anM)) are given by polynomials in n for large n.