The cleanness of (symbolic) powers of Stanley-Reisner ideals

Bandari, Somayeh, and Jahan, Ali Soleyman. "The cleanness of (symbolic) powers of Stanley-Reisner ideals." Czechoslovak Mathematical Journal 67.3 (2017): 767-778. <http://eudml.org/doc/294122>.

@article{Bandari2017,
abstract = {Let $\Delta $ be a pure simplicial complex on the vertex set $[n]=\lbrace 1,\ldots ,n\rbrace $ 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 $m\ge 3$. 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.},
author = {Bandari, Somayeh, Jahan, Ali Soleyman},
journal = {Czechoslovak Mathematical Journal},
keywords = {clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power},
language = {eng},
number = {3},
pages = {767-778},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The cleanness of (symbolic) powers of Stanley-Reisner ideals},
url = {http://eudml.org/doc/294122},
volume = {67},
year = {2017},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 767
EP - 778
AB - Let $\Delta $ be a pure simplicial complex on the vertex set $[n]=\lbrace 1,\ldots ,n\rbrace $ 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 $m\ge 3$. 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.
LA - eng
KW - clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power
UR - http://eudml.org/doc/294122
ER -