Graph and depth of a monomial squarefree ideal
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- by Dorin Popescu
- Proc. Amer. Math. Soc. 140 (2012), 3813-3822
- DOI: https://doi.org/10.1090/S0002-9939-2012-11371-1
- Published electronically: March 19, 2012
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Abstract:
Let $I$ be a monomial squarefree ideal of a polynomial ring $S$ over a field $K$ such that the sum of every three different ideals of its minimal prime ideals is the maximal ideal of $S$, or more generally a constant ideal. We associate to $I$ a graph on $[s]$, $s=|\operatorname {Min}S/I|$, on which we may read the depth of $I$. In particular, $\operatorname {depth_S}I$ does not depend on char $K$. Also we show that $I$ satisfies Stanley’s Conjecture.References
- Jürgen Herzog, Marius Vladoiu, and Xinxian Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra 322 (2009), no. 9, 3151–3169. MR 2567414, DOI 10.1016/j.jalgebra.2008.01.006
- J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc. 140 (2012), 493–504.
- M. Ishaq, Upper bounds for the Stanley depth, to appear in Comm. Algebra, arXiv:AC/1003.3471.
- Gennady Lyubeznik, On the arithmetical rank of monomial ideals, J. Algebra 112 (1988), no. 1, 86–89. MR 921965, DOI 10.1016/0021-8693(88)90133-0
- Adrian Popescu, Special Stanley decompositions, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 4, 363–372. MR 2777680
- Dorin Popescu, Stanley depth of multigraded modules, J. Algebra 321 (2009), no. 10, 2782–2797. MR 2512626, DOI 10.1016/j.jalgebra.2009.03.009
- Dorin Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009), no. 3, 377–382. MR 2554495
- D. Popescu, Stanley conjecture on intersections of four monomial prime ideals. Submitted to Comm. Algebra.
- Dorin Popescu and Muhammad Imran Qureshi, Computing the Stanley depth, J. Algebra 323 (2010), no. 10, 2943–2959. MR 2609185, DOI 10.1016/j.jalgebra.2009.11.025
- Asia Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra 38 (2010), no. 2, 773–784. MR 2598911, DOI 10.1080/00927870902829056
- Richard P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. MR 666158, DOI 10.1007/BF01394054
- Rafael H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR 1800904
Bibliographic Information
- Dorin Popescu
- Affiliation: Institute of Mathematics “Simion Stoilow”, University of Bucharest, P.O. Box 1-764, Bucharest 014700, Romania
- Email: dorin.popescu@imar.ro
- Received by editor(s): May 6, 2011
- Published electronically: March 19, 2012
- Additional Notes: The support from CNCSIS grant PN II-542/2009 of the Romanian Ministry of Education, Research and Innovation is gratefully acknowledged.
- Communicated by: Irena Peeva
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3813-3822
- MSC (2010): Primary 13C15; Secondary 13F20, 05E40, 13F55, 05C25
- DOI: https://doi.org/10.1090/S0002-9939-2012-11371-1
- MathSciNet review: 2944722