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Stanley depth of factors of polymatroidal ideals and the edge ideal of forests. (English) Zbl 1354.13017

Let \(S=k[x_1,\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(k\) and \(J\subseteq I\subseteq S\) be two polymatroidal ideals. By using the affine ranks of \(I\) and \(J\), the authors provide a lower bound for the Stanley depth of \(I/J\). The affine rank is also used for proving that \(I^k/I^{k+1}\) satisfies the Stanley’s inequality, that is \(\mathrm{sdepth}(I^k/I^{k+1})\geq\mathrm{depth}(I^k/I^{k+1})\), for \(k\gg0\). Moreover they show that \(I^k/I^{k+1}\), \(k\gg0\), satisfies the Stanley’s inequality, when \(I\) is the edge ideals of a forest graph with \(p\) connected components.

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
05E99 Algebraic combinatorics
13C13 Other special types of modules and ideals in commutative rings
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[1] Bandari, S.; Herzog, J., Monomial localizations and polymatroidal ideals, European J. Combin., 34, 752-763, (2013) · Zbl 1273.13035
[2] Cimpoeas, M., Stanley depth of quotient of monomial complete intersection ideals, Comm. Algebra, 42, 4274-4280, (2014) · Zbl 1329.13046
[3] A. M. Duval et al., A non-partitionable Cohen-Macaulay simplicial complex, preprint. · Zbl 1341.05256
[4] J. Herzog, A survey on Stanley depth. In “Monomial Ideals, Computations and Applications”, A. Bigatti, P. Gimenez, E. Saenez-de-Cabezon (Eds.), Proceedings of a MONICA 2011. Lecture Notes in Math. 2083, Springer (2013). · Zbl 1310.13001
[5] Herzog, J.; Hibi, T., Discrete polymatroids, J. Algebraic Combin., 16, 239-268, (2002) · Zbl 1012.05046
[6] J. Herzog and T. Hibi, Monomial Ideals, Springer-Verlag, 2011. · Zbl 1258.13014
[7] Herzog, J.; Hibi, T., The depth of powers of an ideal, J. Algebra, 291, 325-650, (2005) · Zbl 1096.13015
[8] Herzog, J.; Rauf, A.; Vladoiu, M., The stable set of associated prime ideals of a polymatroidal ideals, J. Algebraic Combin., 37, 289-312, (2013) · Zbl 1258.13014
[9] Herzog, J.; Vladoiu, M.; Zheng, X., How to compute the Stanley depth of a monomial ideal, J. Algebra, 322, 3151-3169, (2009) · Zbl 1186.13019
[10] Morey, S., Depth of powers of the edge ideal of a tree, Comm. Algebra, 38, 4042-4055, (2010) · Zbl 1210.13020
[11] Mohammadi, F.; Moradi, S., Weakly polymatroidal ideals with applications to vertex ideals, Osaka J. Math., 47, 627-636, (2010) · Zbl 1203.13017
[12] M. R. Pournaki et al., What is \({…}\) Stanley depth? Notices Amer. Math. Soc. 56 (2009), 1106-1108. · Zbl 1177.13056
[13] M. R. Pournaki, S. A. Seyed Fakhari and S. Yassemi, Stanley depth of powers of the edge ideal of a forest, Proc. Amer. Math. Soc. 141 (2013), 3327-3336. · Zbl 1303.13014
[14] M. R. Pournaki, S. A. Seyed Fakhari and S. Yassemi, On the Stanley depth of weakly polymatroidal ideals, Arch. Math. (Basel) 100 (2013), 115-121. · Zbl 1261.13006
[15] Seyed Fakhari, S. A., Stanley depth of weakly polymatroidal ideals and squarefree monomial ideals, Illinois J. Math., 57, 871-881, (2013) · Zbl 1303.13014
[16] S. A. Seyed Fakhari, Stanley depth of weakly polymatroidal ideals, Arch. Math. (Basel), 103 (2014), 229-233. · Zbl 1308.13017
[17] Soleyman Jahan, A., Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality, Manuscripta. Math., 130, 533-550, (2009) · Zbl 1183.13013
[18] Stanley, R. P., Linear Diophantine equations and local cohomology, Invent. Math., 68, 175-193, (1982) · Zbl 0516.10009
[19] W. Vasconcelos, Integral Closure, Rees Algebras, Multiplicities, Algorithms, Springer Monographs in Mathematics. Springer-verlag, Berlin, 2005. · Zbl 1082.13006
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