The order of dominance of a monomial ideal

Abstract

Let S be a polynomial ring innvariables over a field, and consider a monomial ideal M=(m1, . . . , mq) of S. We introduce a new invariant, called order of dominance of S/M, and denoted odom (S/M), which has many similarities with the codimension of S/M. We use the order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. In addition, we show that odom (S/M) has the following properties: (i) codim(S/M)⩽odom(S/M)⩽pd(S/M). (ii) pd(S/M)=n if and only if odom(S/M)=n. (iii) pd(S/M)=1 if and only if odom(S/M)=1. (iv) If odom(S/M)=n−1, then pd(S/M)=n−1. (v) If odom(S/M)=q−1, then pd(S/M)=q−1. (vi) If n=3, then pd(S/M)=odom(S/M).