### The order of dominance of a monomial ideal

G. Alesandroni

#### 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 $$\operatorname{pd}(S/M)=\operatorname{odom}(S/M)$$.

#### Keywords

monomial ideal, codimension, projective dimension, Betti number

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