The Evans-Griffith Syzygy Theorem and Bass Numbers

Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring containing a field. The syzygy theorem of Evans and Griffith (see The syzygy problem, Ann. of Math. (2) 114 (1981), 323-353) says that a nonfree $m$th syzygy module $M$ over $R$ which has finite projective dimension must have rank $\geq m$. This theorem is an assertion about the ranks of the homomorphisms in certain acyclic complexes. It is the aim of this paper to demonstrate that the condition of acyclicity can be relaxed in a natural way. We shall use the generalization thus obtained to show that the Bass numbers of a module satisfy restrictions analogous to those which the syzygy theorem imposes on Betti numbers.