Symbolic powers and matroids
Author:
Matteo Varbaro
Journal:
Proc. Amer. Math. Soc. 139 (2011), 2357-2366
MSC (2010):
Primary 13A15, 05E45; Secondary 13A30
DOI:
https://doi.org/10.1090/S0002-9939-2010-10685-8
Published electronically:
December 8, 2010
MathSciNet review:
2784800
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that all the symbolic powers of a Stanley-Reisner ideal are Cohen-Macaulay if and only if the simplicial complex
is a matroid.
- [BCV] Bruno Benedetti, Alexandru Constantinescu, and Matteo Varbaro, Dimension, depth and zero-divisors of the algebra of basic 𝐾-covers of a graph, Matematiche (Catania) 63 (2008), no. 2, 117–156 (2009). MR 2531656
- [BH] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- [BrVe] Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963
- [CV] A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebra, available online at arXiv:1004.4980v1, 2010.
- [HHT] Jürgen Herzog, Takayuki Hibi, and Ngô Viêt Trung, Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322. MR 2298826, https://doi.org/10.1016/j.aim.2006.06.007
- [Ly] Gennady Lyubeznik, On the local cohomology modules 𝐻ⁱ_{𝔞}(ℜ) for ideals 𝔞 generated by monomials in an ℜ-sequence, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 214–220. MR 775884, https://doi.org/10.1007/BFb0099364
- [MS] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
- [MT] N. C. Minh, N. V. Trung, Cohen-Macaulayness of powers of monomial ideals and symbolic powers of Stanley-Reisner ideals, available online at arXiv:1003.2152v1, 2010.
- [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, https://doi.org/10.1017/s0305004100029194
- [Ox] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- [St] Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- [TY] Naoki Terai and Ken-Ichi Yoshida, Locally complete intersection Stanley-Reisner ideals, Illinois J. Math. 53 (2009), no. 2, 413–429. MR 2594636
- [We] D. J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. MR 0427112
- [Ya] Zhao Yan, An étale analog of the Goresky-MacPherson formula for subspace arrangements, J. Pure Appl. Algebra 146 (2000), no. 3, 305–318. MR 1742346, https://doi.org/10.1016/S0022-4049(98)00128-5
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Additional Information
Matteo Varbaro
Affiliation:
Dipartimento di Matematica, Università degli Studi di Genova, Via Dodrcaneso 35, 16145, Genova, Italy
Email:
varbaro@dima.unige.it
DOI:
https://doi.org/10.1090/S0002-9939-2010-10685-8
Received by editor(s):
March 14, 2010
Received by editor(s) in revised form:
June 25, 2010
Published electronically:
December 8, 2010
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.