The order of dominance of a monomial ideal

TL;DR

This paper introduces the order of dominance, a new invariant for monomial ideals, and uses it to characterize Cohen-Macaulay Scarf ideals and minimal Taylor resolutions, revealing relationships with projective dimension.

Contribution

The paper defines the order of dominance for monomial ideals and applies it to characterize Cohen-Macaulay Scarf ideals and minimal Taylor resolutions.

Findings

odom(S/M) is bounded between codimension and projective dimension.

odom(S/M) equals the projective dimension when it is n.

odom(S/M) equals 1 if and only if the projective dimension is 1.

Abstract

Let S be a polynomial ring in n variables over a field, and let M be a monomial ideal of S. We introduce a new invariant, called the order of dominance of S/M, denoted odom(S/M), which has many similarities with the codimension of S/M. We use this order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. We also 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.