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Let (R, 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 mth syzygy module M over R which has finite projective dimension must have rank > 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. The acyclicity criterion of Buchsbaum and Eisenbud [3] is an essential tool in what follows. Theorem 1 below is the general version given by Northcott [14] which uses the polynomial or true grade of a module M with respect to an ideal I (see [14]); it is denoted by Grade(I, M). It is unnecessary for our considerations to know the exact definition of Grade(I, M). We will only need the inequality Grade(I, M) > grade(I, M) where grade(I, M) denotes the "classical" grade of an ideal I with respect to a module M; furthermore one should note that Grade(I, M) = grade(I, M) if M is a finite (i.e., finitely generated) module over a Noetherian ring R [14]. Other notation to be explained: if (: F -? G is a homomorphism of finite free R-modules, then Iu(Q) denotes the ideal generated by the u-minors of (a matrix) of (O. Furthermore, rank(Q, M) is the largest integer v for which I, (qo)M =$ 0. Several times we shall use that for a prime ideal p one has Iu (() t p if and only if Im(Q 0 Rp) contains a free direct Rp-summand of rank u of Gp; this is an easy consequence of Nakayama's lemma. Theorem 1. Let R be an arbitrary commutative ring, F. :0-)Fs Fs F_ (p F1 Fo a complex offinitefree R-modules, and M =$ 0 an R-module. For i = 1,..., s we set ri = Z>=i(1)J-i rankFj. (a) If F. X M is acyclic, then (i) rank(pi, M) = ri for i = 1,..., s, and (ii) Grade(Ir, (Q), M) > i for i = 1, .. ., s . (b) Conversely, if condition (a) (ii) is satisfied, then F. ? M is acyclic. Received by the editors November 1, 1990 and, in revised form, January 28, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 3C05, 1 3D25. (? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

The Evans-Griffith syzygy theorem and Bass numbers