## Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals

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- by S. A. Seyed Fakhari PDF
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## 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.## References

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## Additional Information

**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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 1849-1862 - MSC (2010): Primary 13C15, 05E99; Secondary 13C13
- DOI: https://doi.org/10.1090/proc/14864
- MathSciNet review: 4078072