The order of dominance of a monomial ideal

Abstract

Let \(S\) be a polynomial ring in \(n\) variables over a field, and consider a monomial ideal \(M=(m_1,\ldots,m_q)\) of \(S\). We introduce a new invariant, called order of dominance of \(S/M\), and denoted \(\operatorname{odom}(S/M)\), 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 \(\operatorname{odom}(S/M)\) has the following properties:

(i) \(\operatorname{codim}(S/M) \leq \operatorname{odom}(S/M)\leq \operatorname{pd}(S/M)\).

(ii) \(\operatorname{pd}(S/M)=n\) if and only if \(\operatorname{odom}(S/M)=n\).

(iii) \(\operatorname{pd}(S/M)=1\) if and only if \(\operatorname{odom}(S/M)=1\).

(iv) If \(\operatorname{odom}(S/M)=n-1\), then \(\operatorname{pd}(S/M)=n-1\).

(v) If \(\operatorname{odom}(S/M)=q-1\), then \(\operatorname{pd}(S/M)=q-1\).

(vi) If \(n=3\), then \(\operatorname{pd}(S/M)=\operatorname{odom}(S/M)\).