Stanley depth and size of a monomial ideal

Authors:
Jürgen Herzog, Dorin Popescu and Marius Vladoiu

Journal:
Proc. Amer. Math. Soc. **140** (2012), 493-504

MSC (2010):
Primary 13C15; Secondary 13P10, 13F55, 13F20

DOI:
https://doi.org/10.1090/S0002-9939-2011-11160-2

Published electronically:
July 5, 2011

MathSciNet review:
2846317

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by is a lower bound for its depth. We show that the size increased by is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.

**1.**W. Bruns and J. Herzog,*Cohen-Macaulay rings*. Revised edition. Cambridge University Press (1998). MR**1251956 (95h:13020)****2.**M. Cimpoeaş, Some remarks on the Stanley depth for multigraded modules. Le Matematiche**LXIII**(2008), 165-171 (2009). MR**2531658 (2010d:13032)****3.**J. Herzog and T. Hibi,*Monomial Ideals*. Graduate Texts in Math. 260. Springer-Verlag, 2011. MR**2724673****4.**J. Herzog, M. Vladoiu and X. Zheng, How to compute the Stanley depth of a monomial ideal. J. Algebra**322**, 3151-3169 (2009). MR**2567414 (2010k:13036)****5.**G. Lyubeznik, On the arithmetical rank of monomial ideals. J. Algebra**112**, 86-89 (1988). MR**921965 (89b:13020)****6.**A. Popescu, Special Stanley Decompositions. Bull. Math. Soc. Sci. Math. Roumanie**53**(101), 361-372 (2010).**7.**D. Popescu, Stanley conjecture on intersections of four monomial prime ideals. arXiv.AC/1009.5646.**8.**A. Rauf, Depth and Stanley Depth of Multigraded Modules. Comm. in Algebra**38**, 773-784 (2010). MR**2598911****9.**T. Römer, Generalized Alexander duality and applications. Osaka J. Math.**38**, 469-485 (2001). MR**1833633 (2002c:13029)****10.**Y. Shen, Stanley depth of complete intersection monomial ideals and upper-discrete partitions. J. Algebra**321**, 1285-1292 (2009). MR**2489900 (2009k:13044)****11.**A. Soleyman Jahan, Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality. Manuscripta Math.**130**, 533-550 (2009). MR**2563149 (2010j:13042)****12.**K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree -graded modules. J. Algebra**225**, 630-645 (2000). MR**1741555 (2000m:13036)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
13C15,
13P10,
13F55,
13F20

Retrieve articles in all journals with MSC (2010): 13C15, 13P10, 13F55, 13F20

Additional Information

**Jürgen Herzog**

Affiliation:
Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany

Email:
juergen.herzog@uni-essen.de

**Dorin Popescu**

Affiliation:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy, University of Bucharest, P. O. Box 1-764, Bucharest 014700, Romania

Email:
dorin.popescu@imar.ro

**Marius Vladoiu**

Affiliation:
Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, Bucharest, RO-010014, Romania

Email:
vladoiu@gta.math.unibuc.ro

DOI:
https://doi.org/10.1090/S0002-9939-2011-11160-2

Received by editor(s):
November 30, 2010

Published electronically:
July 5, 2011

Additional Notes:
This paper was partially written during the visit of the first author at the Institute of Mathematics “Simion Stoilow” of the Romanian Academy supported by a BitDefender Invited Professor Scholarship, 2010

The second and third authors were partially supported by the CNCSIS grant PN II-542/2009, respectively CNCSIS grant TE$_46$ no. 83/2010, of the Romanian Ministry of Education, Research and Innovation. They also want to express their gratitude to ASSMS of GC University Lahore for creating a very appropriate atmosphere for research work.

Communicated by:
Irena Peeva

Article copyright:
© Copyright 2011
American Mathematical Society