Open Access Open Access  Restricted Access Subscription Access

Stanley depth and symbolic powers of monomial ideals

S. A. Seyed Fakhari

Abstract


The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities $\mathrm{sdepth} (S/I^{(ks)}) \leq \mathrm{sdepth} (S/I^{(s)})$ and $\mathrm{sdepth}(I^{(ks)}) \leq \mathrm{sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, $\mathrm{sdepth}(I^{(k+d)})\leq \mathrm{sdepth}(I^{{(k)}})$ and $\mathrm{sdepth}(S/I^{(k+d)})\leq \mathrm{sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.


Full Text:

PDF

References


Apel, J., On a conjecture of R. P. Stanley. II. Quotients modulo monomial ideals, J. Algebraic Combin. 17 (2003), no. 1, 57–74. http://dx.doi.org/10.1023/A:1021916908512

Cimpoeaş, M., Several inequalities regarding Stanley depth, Rom. J. Math. Comput. Sci. 2 (2012), no. 1, 28–40.

Herzog, J., A survey on Stanley depth, Monomial ideals, computations and applications, Lecture Notes in Math., vol. 2083, Springer, Heidelberg, 2013, pp. 3--45. http://dx.doi.org/10.1007/978-3-642-38742-5_1

Herzog, J. and Hibi, T., Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. http://dx.doi.org/10.1007/978-0-85729-106-6

Herzog, J., Hibi, T., and Trung, N. V., Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322. http://dx.doi.org/10.1016/j.aim.2006.06.007

Ishaq, M., Upper bounds for the Stanley depth, Comm. Algebra 40 (2012), no. 1, 87–97. http://dx.doi.org/10.1080/00927872.2010.523642

Popescu, D., Bounds of Stanley depth, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 19 (2011), no. 2, 187–194.

Pournaki, M. R., Seyed Fakhari, S. A., Tousi, M., and Yassemi, S., What is $dots $ Stanley depth?, Notices Amer. Math. Soc. 56 (2009), no. 9, 1106–1108.

Seyed Fakhari, S. A., Stanley depth of the integral closure of monomial ideals, Collect. Math. 64 (2013), no. 3, 351–362. http://dx.doi.org/10.1007/s13348-012-0077-9

Stanley, R. P., Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. http://dx.doi.org/10.1007/BF01394054




DOI: http://dx.doi.org/10.7146/math.scand.a-25501

Refbacks

  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.
OK


ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library