Increasing sequences of Betti numbers

Abstract

We study the sequence of Betti numbers {βt(M)}fci of an arbitrary finitely generated nonfree module M over a commutative noetherian local ring R and show that for a certain class of rings this sequence is always nondecreasing, while for a certain subclass of rings, the subsequence 2 is strictly increasing. In [3], a class of commutative noetherian local rings (R, m) called BNSI rings was introduced. These rings have the property that for every finitely generated nonfree module M, the sequence of Betti numbers {frCM)},^ is strictly increasing. Recall that βi(M) is the dimension of the R/m-vector space Torf (M, R/m); equivalents, it is the rank of the free module Ft where >F, >F^ > >F0 >M ,0 is a minimal R-free resolution of M. A class of BNSI rings was given in [3, Theorem 3.2A]: Let (S, tt) be a noetherian local ring and let J be an ideal which is not contained in any prime ideal of grade 1. If & is a domain, then S/xtJ is a BNSI ring. In this note, using a result of G. Levin [2] we prove: THEOREM 1.1. Let (S, n) be a noetherian local ring of Krull dimension d ^ 2. Then for n sufficiently large, the local ring (R, m) = (S/nn, n/n%) has the property that for all finitely generated nonfree R-modules M, the sequence {βi(M)}^2 is strictly increasing. In fact, for all i ^ 2, βi+1(M) - βt(M) ^ d - 1.