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arXiv · 1306.0542

Stanley depth and symbolic powers of monomial ideals

S. A. Seyed Fakhari
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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 ${\rm sdepth} (S/I^{(ks)}) \leq {\rm sdepth} (S/I^{(s)})$ and ${\rm sdepth} (I^{(ks)}) \leq {\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, ${\rm sdepth}(I^{(k+d)})\leq {\rm sdepth}(I^{{(k)}})$ and ${\rm sdepth}(S/I^{(k+d)})\leq {\rm sdepth}(S/I^{{(k)}})$. 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.

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S. A. Seyed Fakhari. 2013-06-03. Stanley depth and symbolic powers of monomial ideals. https://arxiv.org/abs/1306.0542

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