The cleanness of (symbolic) powers of Stanley-Reisner ideals
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 767-778
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Let $\Delta $ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots ,n\}$ and $I_\Delta $ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots ,x_n]$. We show that $\Delta $ is a matroid (complete intersection) if and only if $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$) is clean for all $m\in \mathbb {N}$ and this is equivalent to saying that $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb {N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq 3$. If $\dim (\Delta )=1$, we also prove that $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$) is clean if and only if $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$, respectively) is Cohen-Macaulay.
Let $\Delta $ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots ,n\}$ and $I_\Delta $ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots ,x_n]$. We show that $\Delta $ is a matroid (complete intersection) if and only if $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$) is clean for all $m\in \mathbb {N}$ and this is equivalent to saying that $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb {N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq 3$. If $\dim (\Delta )=1$, we also prove that $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$) is clean if and only if $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$, respectively) is Cohen-Macaulay.
DOI :
10.21136/CMJ.2017.0173-16
Classification :
05E40, 13F20, 13F55
Keywords: clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power
Keywords: clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power
@article{10_21136_CMJ_2017_0173_16,
author = {Bandari, Somayeh and Jahan, Ali Soleyman},
title = {The cleanness of (symbolic) powers of {Stanley-Reisner} ideals},
journal = {Czechoslovak Mathematical Journal},
pages = {767--778},
year = {2017},
volume = {67},
number = {3},
doi = {10.21136/CMJ.2017.0173-16},
mrnumber = {3697915},
zbl = {06770129},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0173-16/}
}
TY - JOUR AU - Bandari, Somayeh AU - Jahan, Ali Soleyman TI - The cleanness of (symbolic) powers of Stanley-Reisner ideals JO - Czechoslovak Mathematical Journal PY - 2017 SP - 767 EP - 778 VL - 67 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0173-16/ DO - 10.21136/CMJ.2017.0173-16 LA - en ID - 10_21136_CMJ_2017_0173_16 ER -
%0 Journal Article %A Bandari, Somayeh %A Jahan, Ali Soleyman %T The cleanness of (symbolic) powers of Stanley-Reisner ideals %J Czechoslovak Mathematical Journal %D 2017 %P 767-778 %V 67 %N 3 %U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0173-16/ %R 10.21136/CMJ.2017.0173-16 %G en %F 10_21136_CMJ_2017_0173_16
Bandari, Somayeh; Jahan, Ali Soleyman. The cleanness of (symbolic) powers of Stanley-Reisner ideals. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 767-778. doi: 10.21136/CMJ.2017.0173-16
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