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