Symbolic powers and matroids

Varbaro, Matteo

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

Symbolic powers and matroids
HTML articles powered by AMS MathViewer

by Matteo Varbaro PDF

Proc. Amer. Math. Soc. 139 (2011), 2357-2366 Request permission

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.

References

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
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
A. Constantinescu, M. Varbaro, Koszulness, Krull dimension and other properties of graph-related algebra, available online at arXiv:1004.4980v1, 2010.
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, DOI 10.1016/j.aim.2006.06.007
Gennady Lyubeznik, On the local cohomology modules $H^i_{{\mathfrak {a}}}(R)$ for ideals ${\mathfrak {a}}$ 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, DOI 10.1007/BFb0099364
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
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.
D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, DOI 10.1017/s0305004100029194
James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Naoki Terai and Ken-Ichi Yoshida, Locally complete intersection Stanley-Reisner ideals, Illinois J. Math. 53 (2009), no. 2, 413–429. MR 2594636
D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112
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, DOI 10.1016/S0022-4049(98)00128-5