Some results on the local cohomology of minimax modules
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 327-333
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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 $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ 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 $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i$, 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 $i$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.
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 $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ 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 $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i$, 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 $i$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.
DOI :
10.1007/s10587-014-0104-y
Classification :
13C05, 13D45, 13E10
Keywords: local cohomology module; Krull dimension; minimax module; cofinite module; weakly Laskerian module; associated primes
Keywords: local cohomology module; Krull dimension; minimax module; cofinite module; weakly Laskerian module; associated primes
@article{10_1007_s10587_014_0104_y,
author = {Abbasi, Ahmad and Roshan-Shekalgourabi, Hajar and Hassanzadeh-Lelekaami, Dawood},
title = {Some results on the local cohomology of minimax modules},
journal = {Czechoslovak Mathematical Journal},
pages = {327--333},
year = {2014},
volume = {64},
number = {2},
doi = {10.1007/s10587-014-0104-y},
mrnumber = {3277739},
zbl = {06391497},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0104-y/}
}
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 SP - 327 EP - 333 VL - 64 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0104-y/ DO - 10.1007/s10587-014-0104-y LA - en ID - 10_1007_s10587_014_0104_y ER -
%0 Journal Article %A Abbasi, Ahmad %A Roshan-Shekalgourabi, Hajar %A Hassanzadeh-Lelekaami, Dawood %T Some results on the local cohomology of minimax modules %J Czechoslovak Mathematical Journal %D 2014 %P 327-333 %V 64 %N 2 %U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0104-y/ %R 10.1007/s10587-014-0104-y %G en %F 10_1007_s10587_014_0104_y
Abbasi, Ahmad; Roshan-Shekalgourabi, Hajar; Hassanzadeh-Lelekaami, Dawood. Some results on the local cohomology of minimax modules. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 327-333. doi: 10.1007/s10587-014-0104-y
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