Pretty cleanness and filter-regular sequences

Bandari, Somayeh, Divaani-Aazar, Kamran, and Jahan, Ali Soleyman. "Pretty cleanness and filter-regular sequences." Czechoslovak Mathematical Journal 64.4 (2014): 933-944. <http://eudml.org/doc/269853>.

@article{Bandari2014,
abstract = {Let $K$ be a field and $S=K[x_1,\ldots , x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots , u_r$ be monomials in $S$. We prove that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots , u_r)$ is pretty clean. Also, we show that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then Stanley’s conjecture is true for $S/I$ if and only if it is true for $S/(I,u_1, \ldots , u_r)$. Finally, we prove that if $u_1,\ldots , u_r$ is a minimal set of generators for $I$ which form either a $d$-sequence, proper sequence or strong $s$-sequence (with respect to the reverse lexicographic order), then $S/I$ is pretty clean.},
author = {Bandari, Somayeh, Divaani-Aazar, Kamran, Jahan, Ali Soleyman},
journal = {Czechoslovak Mathematical Journal},
keywords = {almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module; almost clean module; clean module; -sequence; filter-regular sequence; pretty clean module},
language = {eng},
number = {4},
pages = {933-944},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pretty cleanness and filter-regular sequences},
url = {http://eudml.org/doc/269853},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bandari, Somayeh
AU - Divaani-Aazar, Kamran
AU - Jahan, Ali Soleyman
TI - Pretty cleanness and filter-regular sequences
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 933
EP - 944
AB - Let $K$ be a field and $S=K[x_1,\ldots , x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots , u_r$ be monomials in $S$. We prove that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots , u_r)$ is pretty clean. Also, we show that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then Stanley’s conjecture is true for $S/I$ if and only if it is true for $S/(I,u_1, \ldots , u_r)$. Finally, we prove that if $u_1,\ldots , u_r$ is a minimal set of generators for $I$ which form either a $d$-sequence, proper sequence or strong $s$-sequence (with respect to the reverse lexicographic order), then $S/I$ is pretty clean.
LA - eng
KW - almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module; almost clean module; clean module; -sequence; filter-regular sequence; pretty clean module
UR - http://eudml.org/doc/269853
ER -