$F_{\Delta }$ type free resolutions of monomial ideals

Abstract:

Let $M = (m_1, \ldots , m_r)$ be a monomial ideal of $S = k[x_1, \ldots , x_n]$. Bayer-Peeva-Sturmfels studied a subcomplex $F_{\Delta }$ of the Taylor resolution, defined by a simplicial complex $\Delta \subset 2^r$. They proved that if $M$ is generic (i.e., no variable $x_i$ appears with the same non-zero exponent in two distinct monomials which are minimal generators), then $F_{\Delta _M}$ is the minimal free resolution of $S/M$, where $\Delta _M$ is the Scarf complex of $M$. In this paper, we prove the following: for a generic (in the above sense) monomial ideal $M$ and each integer $\operatorname {depth} S/M \leq i < \dim S/M$, there is an embedded prime $P \in \operatorname {Ass} (S/M)$ of $\dim S/P =i$. Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, $\Delta _M$ is shellable). We also study a non-generic monomial ideal $M$ whose minimal free resolution is $F_{\Delta }$ for some $\Delta$. In particular, we prove that if all associated primes of $M$ have the same height, then $M$ is Cohen-Macaulay and $\Delta$ is pure and strongly connected.