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

Author: Kohji Yanagawa
Journal: Proc. Amer. Math. Soc. 127 (1999), 377-383
MSC (1991): Primary 13D02, 13D03, 13H10
MathSciNet review: 1610805
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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.

References [Enhancements On Off] (What's this?)

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Additional Information

Kohji Yanagawa
Affiliation: Graduate School of Science, Osaka University, Toyonaka, Osaka 560, Japan

Keywords: Monomial ideal, generic monomial ideal, Taylor complex, Scarf complex, primary decomposition, minimal free resolution, Cohen-Macaulay ring
Received by editor(s): May 31, 1997
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society

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