Cominimaxness of local cohomology modules

Aghapournahr, Moharram

Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 75-86.

Résumé

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in \mathbb {N}_0$ be an integer and $M$ an $R$-module such that ${\rm Ext}^i_R(R/I,M)$ is minimax for all $i\leq t+1$. We prove that if $H^{i}_{I}(M)$ is ${\rm FD}_{\leq 1}$ (or weakly Laskerian) for all $i$, then the $R$-modules $H^{i}_{I}(M)$ are $I$-cominimax for all $i$ and ${\rm Ext}^i_R(R/I,H^{t}_{I}(M))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\rm Ext}^j_R(N,H^{i}_{I}(M))$ and ${\rm Tor}^R_{j}(N,H^{i}_I(M))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^{i}_{I}(M)$ is ${\rm FD}_{\leq 1}$ (or weakly Laskerian) for all $i$.

Détail

DOI : 10.21136/CMJ.2018.0161-17

Classification : 13C05, 13D45, 13E10
Keywords: local cohomology; ${\rm FD}_{\leq n}$ modules; cofinite modules; cominimax modules