A note on the Hilbert functions of certain ideals which are defined by matrices

Abstract

Let I be a homogeneous ideal in the polynomial ring R = Λ[X 1 , X 2 , …, X N ] , where Λ is a field or, more generally, an Artin ring†. Then R|I has an induced structure as a graded R-module and its homogeneous elements of degree n form a Λ-module of finite length. If this length is denoted by H( n , R|I) , then H( n , R|I) , considered as a function of n , is often known as the Hilbert function of the ideal I although, in other contexts, it is called the Hilbert function of the graded module R|I . We shall adopt the latter terminology.