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 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 of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
- 3. Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. MR 1333388, https://doi.org/10.1090/S0002-9947-96-01534-6
- 4. Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- 5. Mitsuhiro Miyazaki, On the canonical map to the local cohomology of a Stanley-Reisner ring, Bull. Kyoto Univ. Ed. Ser. B 79 (1991), 1–8. MR 1143877
- 6. R. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser (1996). CMP 97:14
- D. Bayer, I. Peeva and B. Sturmfels, Monomial resolutions, Math. Res. Lett. (to appear) (alg-geom/9610012).
- A. Björner, Topological methods, Handbook for combinatorics, 1819-1872, Elsevier, Amsterdam, 1995. MR 1373690
- A. Björner and M. Wachs, Shellable nonpure complex and posets, Trans. Amer. Math. Soc. 348 (1996), 1299-1327. MR 1333388
- W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge University Press, 1993. MR 1251956
- 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 1143877
- R. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser (1996). CMP 97:14
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