Some results on the local cohomology of minimax modules

Abbasi, Ahmad ; Roshan-Shekalgourabi, Hajar ; Hassanzadeh-Lelekaami, Dawood

Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 327-333.

Résumé

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.

Détail

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