Cohen-Macaulay local rings of dimension two and an extended version of a conjecture of J. Sally

Abstract

In this paper we prove an extended version of a conjecture of J. Sally. Let ( A, M) be a Cohen-Macaulay local ring of dimension d, multiplicity e and embedding codimension h. If the initial degree of A is bigger than or equal to t and e= h+t−1 h +1 , we prove that the depth of the associated graded ring of A is at least d − 1 and the h-vector of A has no negative components. The conjecture of Sally was dealing with the case t = 2 and was proved by these authors in a previous paper. Some new formulas relating certain numerical characters of a two-dimensional Cohen-Macaulay local ring are also given.