Complete intersections and Gorenstein ideals

Abstract

All rings considered in this paper are noetherian commutative K-algebras, where K is a fixed field. A ring R is a Gorenstein ring if the injective dimension of R is finite. In a ring R, we say an ideal I is Gorenstein (respectively, zero dimensional) if R/I is Gorenstein (resp., zero dimensional). We say an ideal I is a complete intersection in R if I can be generated by an R-sequence. These concepts are related in that if an ideal I in a Gorenstein ring R is a zero dimensional complete intersection then it is also a zero dimensional Gorenstein ideal [3]. Both types of ideals play important roles in both algebraic geometry and commutative ring theory. The underlying goal of this paper is to demonstrate the utility of studying these ideals from a different point of view, one which has basically been overlooked. We are led to employ techniques which use only the tools of linear algebra and hence have the advantage of being algorithmic. The first main result of the paper is to give a simple method for determining when a zero dimensional Gorenstein ideal in a polynomial ring over k is in fact a complete intersection. Secondly, restricting our attention to homogeneous ideals, we show that if I is a homogeneous zero dimensional complete intersection, I must contain a polynomial of “small” degree; whereas this need not be true of homogeneous zero dimensional Gorenstein ideals.