type free resolutions of monomial ideals

Author:
Kohji Yanagawa

Journal:
Proc. Amer. Math. Soc. **127** (1999), 377-383

MSC (1991):
Primary 13D02, 13D03, 13H10

DOI:
https://doi.org/10.1090/S0002-9939-99-04947-3

MathSciNet review:
1610805

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is *generic* (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the *minimal* free resolution of , where is the *Scarf complex* of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an *embedded* prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose *minimal* free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.

**1.**D. Bayer, I. Peeva and B. Sturmfels,*Monomial resolutions*, Math. Res. Lett. (to appear) (`alg-geom/9610012`).**2.**A. Björner,*Topological methods*, Handbook for combinatorics, 1819-1872, Elsevier, Amsterdam, 1995. MR**96m:52012****3.**A. Björner and M. Wachs,*Shellable nonpure complex and posets*, Trans. Amer. Math. Soc.**348**(1996), 1299-1327. MR**96i:06008****4.**W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge University Press, 1993. MR**95h:13020****5.**M. Miyazaki,*On the canonical map to the local cohomology of a Stanley-Reisner ring*, Bulletin of Kyoto University of Education Ser. B79, 1-8 (1991). MR**93b:13040****6.**R. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser (1996). CMP**97:14**

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Additional Information

**Kohji Yanagawa**

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

Email:
yanagawa@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-99-04947-3

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