Pretty cleanness and filter-regular sequences
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 933-944
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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.
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.
DOI :
10.1007/s10587-014-0144-3
Classification :
05E40, 13F20
Keywords: almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module
Keywords: almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module
@article{10_1007_s10587_014_0144_3,
author = {Bandari, Somayeh and Divaani-Aazar, Kamran and Jahan, Ali Soleyman},
title = {Pretty cleanness and filter-regular sequences},
journal = {Czechoslovak Mathematical Journal},
pages = {933--944},
year = {2014},
volume = {64},
number = {4},
doi = {10.1007/s10587-014-0144-3},
mrnumber = {3304789},
zbl = {06433705},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0144-3/}
}
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 SP - 933 EP - 944 VL - 64 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0144-3/ DO - 10.1007/s10587-014-0144-3 LA - en ID - 10_1007_s10587_014_0144_3 ER -
%0 Journal Article %A Bandari, Somayeh %A Divaani-Aazar, Kamran %A Jahan, Ali Soleyman %T Pretty cleanness and filter-regular sequences %J Czechoslovak Mathematical Journal %D 2014 %P 933-944 %V 64 %N 4 %U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0144-3/ %R 10.1007/s10587-014-0144-3 %G en %F 10_1007_s10587_014_0144_3
Bandari, Somayeh; Divaani-Aazar, Kamran; Jahan, Ali Soleyman. Pretty cleanness and filter-regular sequences. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 933-944. doi: 10.1007/s10587-014-0144-3
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