A generalization of the finiteness problem of the local cohomology modules

Abbasi, Ahmad, and Roshan-Shekalgourabi, Hajar. "A generalization of the finiteness problem of the local cohomology modules." Czechoslovak Mathematical Journal 64.1 (2014): 69-78. <http://eudml.org/doc/261979>.

@article{Abbasi2014,
abstract = {Let $R$ be a commutative Noetherian ring and $\{\mathfrak \{a\}\}$ an ideal of $R$. We introduce the concept of $\{\mathfrak \{a\}\}$-weakly Laskerian $R$-modules, and we show that if $M$ is an $\{\mathfrak \{a\}\}$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that $\{\rm Ext\}^j_R(R/\{\mathfrak \{a\}\}, H^i_\{\{\mathfrak \{a\}\}\}(M))$ is $\{\mathfrak \{a\}\}$-weakly Laskerian for all $i<s$ and all $j$, then for any $\{\mathfrak \{a\}\}$-weakly Laskerian submodule $X$ of $H^s_\{\{\mathfrak \{a\}\}\}(M)$, the $R$-module $\{\rm Hom\}_R(R/\{\mathfrak \{a\}\},H^s_\{\{\mathfrak \{a\}\}\}(M)/X)$ is $\{\mathfrak \{a\}\}$-weakly Laskerian. In particular, the set of associated primes of $H^s_\{\mathfrak \{a\}\}(M)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an $\{\mathfrak \{a\}\}$-weakly Laskerian $R$-module such that $ H^i_\{\{\mathfrak \{a\}\}\}(N)$ is $\{\mathfrak \{a\}\}$-weakly Laskerian for all $i<s$, then the set of associated primes of $H^s_\{\mathfrak \{a\}\}(M, N)$ is finite. This generalizes the main result of S. Sohrabi Laleh, M. Y. Sadeghi, and M. Hanifi Mostaghim (2012).},
author = {Abbasi, Ahmad, Roshan-Shekalgourabi, Hajar},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology module; weakly Laskerian module; $\{\mathfrak \{a\}\}$-weakly Laskerian module; associated prime; associated prime; local cohomology module; $\{\mathfrak \{a\}\}$-weakly Laskerian module},
language = {eng},
number = {1},
pages = {69-78},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A generalization of the finiteness problem of the local cohomology modules},
url = {http://eudml.org/doc/261979},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Abbasi, Ahmad
AU - Roshan-Shekalgourabi, Hajar
TI - A generalization of the finiteness problem of the local cohomology modules
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 69
EP - 78
AB - Let $R$ be a commutative Noetherian ring and ${\mathfrak {a}}$ an ideal of $R$. We introduce the concept of ${\mathfrak {a}}$-weakly Laskerian $R$-modules, and we show that if $M$ is an ${\mathfrak {a}}$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\rm Ext}^j_R(R/{\mathfrak {a}}, H^i_{{\mathfrak {a}}}(M))$ is ${\mathfrak {a}}$-weakly Laskerian for all $i<s$ and all $j$, then for any ${\mathfrak {a}}$-weakly Laskerian submodule $X$ of $H^s_{{\mathfrak {a}}}(M)$, the $R$-module ${\rm Hom}_R(R/{\mathfrak {a}},H^s_{{\mathfrak {a}}}(M)/X)$ is ${\mathfrak {a}}$-weakly Laskerian. In particular, the set of associated primes of $H^s_{\mathfrak {a}}(M)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an ${\mathfrak {a}}$-weakly Laskerian $R$-module such that $ H^i_{{\mathfrak {a}}}(N)$ is ${\mathfrak {a}}$-weakly Laskerian for all $i<s$, then the set of associated primes of $H^s_{\mathfrak {a}}(M, N)$ is finite. This generalizes the main result of S. Sohrabi Laleh, M. Y. Sadeghi, and M. Hanifi Mostaghim (2012).
LA - eng
KW - local cohomology module; weakly Laskerian module; ${\mathfrak {a}}$-weakly Laskerian module; associated prime; associated prime; local cohomology module; ${\mathfrak {a}}$-weakly Laskerian module
UR - http://eudml.org/doc/261979
ER -