For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].
@article{bwmeta1.element.doi-10_2478_s11533-009-0037-0,
author = {Mircea Cimpoea\c s},
title = {Stanley depth of monomial ideals with small number of generators},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {629-634},
zbl = {1185.13027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0037-0}
}
Mircea Cimpoeaş. Stanley depth of monomial ideals with small number of generators. Open Mathematics, Tome 7 (2009) pp. 629-634. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0037-0/
[1] Ahmad S., Popescu D., Sequentially Cohen-Macaulay monomial ideals of embedding dimension four, Bull. Math. Soc. Sci. Math. Roumanie, 2007, 50(98), 99–110 | Zbl 1150.13004
[2] Anwar I., Janet’s algorithm, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 11–19
[3] Anwar I., Popescu D., Stanley Conjecture in small embedding dimension, J. Algebra, 2007, 318, 1027–1031 http://dx.doi.org/10.1016/j.jalgebra.2007.06.005[Crossref][WoS] | Zbl 1132.13009
[4] Apel J., On a conjecture of R.P.Stanley, J. Algebraic Combin., 2003, 17, 36–59
[5] Cimpoeas M., Stanley depth for monomial complete intersection, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 205–211 | Zbl 1174.13033
[6] Cimpoeas M., Some remarks on the Stanley depth for multigraded modules, Le Mathematiche, 2008, LXIII, 165–171 | Zbl 1184.13067
[7] Herzog J., Jahan A.S., Yassemi S., Stanley decompositions and partitionable simplicial complexes, J. Algebraic Combin., 2008, 27, 113–125 http://dx.doi.org/10.1007/s10801-007-0076-1[Crossref] | Zbl 1131.13020
[8] Herzog J., Vladoiu M., Zheng X., How to compute the Stanley depth of a monomial ideal, J. Algebra, doi:10:1016/j.jalgebra.2008.01.006, to appear [WoS] | Zbl 1186.13019
[9] Jahan A.S., Prime filtrations of monomial ideals and polarizations, J. Algebra, 2007, 312, 1011–1032 http://dx.doi.org/10.1016/j.jalgebra.2006.11.002[Crossref] | Zbl 1142.13022
[10] Nasir S., Stanley decompositions and localization, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 151–158
[11] Popescu D., Stanley depth of multigraded modules, J. Algebra, 2009, 321(10), 2782–2797 http://dx.doi.org/10.1016/j.jalgebra.2009.03.009[Crossref][WoS] | Zbl 1179.13016
[12] Rauf A., Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Soc. Sci. Math. Roumanie, 2007, 50(98), 347–354 | Zbl 1155.13311
[13] Rauf A., Depth and Stanley depth of multigraded modules, Comm. Algebra, to appear
[14] Shen Y., Stanley depth of complete intersection monomial ideals and upper-discrete partitions, J. Algebra, 2009, 321, 1285–1292 http://dx.doi.org/10.1016/j.jalgebra.2008.11.010[WoS][Crossref] | Zbl 1167.13010