Bounds for the minimum numbers of generators of generalized Cohen-Macaulay ideals

Abstract

Throughout this paper, R denotes a local ring with maxima1 idea1 ,m, Z an ideal of R, and S the factor ring R/Z. We call Z a generalized Cohen-Macaulay idea1 if S is a generalized Cohen-Macaulay ring. By definition [4, (3.1)], a local ring A with maxima1 ideal # is a generalized Cohen-Macaulay ring if the ith local cohomology module H,(A) of A with support b is of finite length for i = O,..., dim A 1 (A is a Cohen-Macaulay ring iff H;(A) = 0 for i = O,..., dim A 1). The class of generalized Cohen-Macaulay local rings is rather large. For instance, if A is a factor ring of a Cohen-Macaulay ring, A being a generalized Cohen-Macaulay ring means that A is equidimensional and A, is a Cohen-Macaulay ring for all prime ideals y+# of A [4, (2.5) and (381. The aim of this paper is to give bounds for the minimum number u(Z) of generators of a generalized Cohen-Macaulay idea1 Z in terms of the following invariants of R and S: m = u(m), n = v(mS), r = dim R, s = dim S, the multiplicities e(R), e(S), and Z(ZYZL~,(S)), i = O,..., s 1, where 1 denotes the length function. Our interest in such bounds for u(Z) arose from the article [7] of Sally; There she proved that if R and S are Cohen-Macaulay rings, then v(Z) < e(R) e(S)r-s-l + r s 1. First, Sally’s result may be generalized as