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


Author: S. A. Seyed Fakhari
Journal: Proc. Amer. Math. Soc. 148 (2020), 1849-1862
MSC (2010): Primary 13C15, 05E99; Secondary 13C13
DOI: https://doi.org/10.1090/proc/14864
Published electronically: December 30, 2019
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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$,

$\displaystyle \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}).$    

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

$\displaystyle \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I),$    

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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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
Email: aminfakhari@ut.ac.ir

DOI: https://doi.org/10.1090/proc/14864
Keywords: Depth, Stanley depth, symbolic power
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
Article copyright: © Copyright 2019 American Mathematical Society