A characterization of cofinite local cohomology modules in a certain Serre class

Abstract

Let $R$ be a commutative Noetherian ring, ${\mathfrak a}$ be an ideal of $R$, $n$ be a non-negative integer, $X$ be an arbitrary $R$-module and $L$ be a finitely generated $R$-module. We characterize when $H^i_{\mathfrak a}(X)$ and $H^i_{\mathfrak a}(L,X)$ are $(FD_{\prec n}, {\mathfrak a})$-cofinite for all $i$, whenever one of the following statements holds: (a) ${\rm ara}({\mathfrak a})\leq 1$, (b) ${\rm dim} R/{\mathfrak a} \leq n+1$, (c) ${\rm dim} R\leq n+2$ or (d) $X$ is an $FD_{\prec n+2}~R$-module. As a consequence, we show that ${\rm Ext}^i_R(L,X)$ is $FD_{\prec n}$ for all $i$ and any ${\frak a}$-torsion finitely generated $R$-module $L$ with ${\rm dim} L\leq n+1$ if one of the above statements holds. Moreover, we obtain that, if $R$ is a semi-local ring with ${\rm dim} R/{\mathfrak a}\leq 2$ and $X$ is an $R$-module such that ${\rm Ext}^j_R(R/{\mathfrak a}, X)$ is $FD_{\prec 1}$ for all $j\leq {\rm dim} X$, then ${\rm Ass}_R(H^i_{\mathfrak a}(L, X))$ is finite for all $i$.