# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Stability of depth and Stanley depth of symbolic powers of squarefree monomial idealsHTML articles powered by AMS MathViewer

by S. A. Seyed Fakhari
Proc. Amer. Math. Soc. 148 (2020), 1849-1862 Request permission

## Abstract:

Let $\mathbb {K}$ be a field and let $S=\mathbb {K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over $\mathbb {K}$. Assume that $I\subset S$ is a squarefree monomial ideal. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers $k, i\geq 1$, \begin{equation*} \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}). \end{equation*} We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence $\{\mathrm {depth}(S/I^{(k)})\}_{k=1}^{\infty }$ is convergent and \begin{equation*} \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I), \end{equation*} where $\ell _s(I)$ denotes the symbolic analytic spread of $I$. We also determine an upper bound for the index of depth stability of symbolic powers of $I$. Next, we consider the Stanley depth of symbolic powers and prove that the sequences $\{\mathrm {sdepth}(S/I^{(k)})\}_{k=1}^{\infty }$ and $\{\mathrm {sdepth}(I^{(k)})\}_{k=1}^{\infty }$ are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.
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• S. A. Seyed Fakhari
• Affiliation: School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran; and Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
• MR Author ID: 881160
• Email: aminfakhari@ut.ac.ir
• Received by editor(s): December 30, 2018
• Received by editor(s) in revised form: August 31, 2019
• Published electronically: December 30, 2019
• Additional Notes: This research was partially funded by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology.
• Communicated by: Claudia Polini