Hilbert-Samuel functions of modules over Cohen-Macaulay rings

Abstract

For a finitely generated, non-free module M over a CM local ring ( R, m, k), it is proved that for n >> 0 the length of TorR 1 ( M, R/m(n+1)) is given by a polynomial of degree dim R-1. The vanishing of Tor(i)(R) ( M, N/m(n+1)N) is studied, with a view towards answering the question: If there exists a finitely generated R-module N with dimN >= 1 such that the projective dimension or the injective dimension of N/m(n+1)N is finite, then is R regular? Upper bounds are provided for n beyond which the question has an affirmative answer.