On ideals of finite homoloǵical dimension in local rings

Abstract

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings. In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n , be described as the set of determinants of matrices obtained by adjoining to a certain ( n –l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q . (This result was established in (3), under the additional condition that Q should be a domain.)