Some results on the local cohomology of minimax modules

Abbasi, Ahmad, Roshan-Shekalgourabi, Hajar, and Hassanzadeh-Lelekaami, Dawood. "Some results on the local cohomology of minimax modules." Czechoslovak Mathematical Journal 64.2 (2014): 327-333. <http://eudml.org/doc/262014>.

@article{Abbasi2014,
abstract = {Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop \{\rm Supp\} H^i_I (M) \le 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop \{\rm Supp\} H^i_I (M)\le 2$ for all $i<t$, then $\{\rm Ext\}^j_R (R/I, H^i_I (M))$ and $\{\rm Hom\}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \ge 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\ge 0$, whenever $\dim R/I \le 2$ and $M$ is weakly Laskerian.},
author = {Abbasi, Ahmad, Roshan-Shekalgourabi, Hajar, Hassanzadeh-Lelekaami, Dawood},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology module; Krull dimension; minimax module; cofinite module; weakly Laskerian module; associated primes; associated prime ideals; cofinite modules; Krull dimension; local cohomology modules; minimax modules; weakly Laskerian modules},
language = {eng},
number = {2},
pages = {327-333},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some results on the local cohomology of minimax modules},
url = {http://eudml.org/doc/262014},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Abbasi, Ahmad
AU - Roshan-Shekalgourabi, Hajar
AU - Hassanzadeh-Lelekaami, Dawood
TI - Some results on the local cohomology of minimax modules
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 327
EP - 333
AB - Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop {\rm Supp} H^i_I (M) \le 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\le 2$ for all $i<t$, then ${\rm Ext}^j_R (R/I, H^i_I (M))$ and ${\rm Hom}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \ge 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\ge 0$, whenever $\dim R/I \le 2$ and $M$ is weakly Laskerian.
LA - eng
KW - local cohomology module; Krull dimension; minimax module; cofinite module; weakly Laskerian module; associated primes; associated prime ideals; cofinite modules; Krull dimension; local cohomology modules; minimax modules; weakly Laskerian modules
UR - http://eudml.org/doc/262014
ER -