Symbolic powers and matroids

Varbaro, Matteo

doi:10.1090/S0002-9939-2010-10685-8

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

Symbolic powers and matroids

Author: Matteo Varbaro
Journal: Proc. Amer. Math. Soc. 139 (2011), 2357-2366
MSC (2010): Primary 13A15, 05E45; Secondary 13A30
Posted: December 8, 2010
MathSciNet review: 2784800
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that all the symbolic powers of a Stanley-Reisner ideal $I_{\Delta}$ are Cohen-Macaulay if and only if the simplicial complex $\Delta$ 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 (2010c:13002)
[BH] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
[BrVe] Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963 (89i:13001)
[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 (2007m:13005), http://dx.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 (86f:14002), http://dx.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 (2006d:13001)
[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 0059889 (15,596a)
[Ox] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1992. MR 1207587 (94d:05033)
[St] Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1996. MR 1453579 (98h:05001)
[TY] Naoki Terai and Ken-Ichi Yoshida, Locally complete intersection Stanley-Reisner ideals, Illinois J. Math. 53 (2009), no. 2, 413–429. MR 2594636 (2011g:13046)
[We] D. J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976. L. M. S. Monographs, No. 8. MR 0427112 (55 #148)
[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 (2000k:14041), http://dx.doi.org/10.1016/S0022-4049(98)00128-5

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13A15, 05E45, 13A30

Retrieve articles in all journals with MSC (2010): 13A15, 05E45, 13A30

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: http://dx.doi.org/10.1090/S0002-9939-2010-10685-8
PII: S 0002-9939(2010)10685-8
Received by editor(s): March 14, 2010
Received by editor(s) in revised form: June 25, 2010
Posted: 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.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|