A criterion for cofiniteness of modules

Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that ${\text{Ext}}_A^i(A\mathrm{/}\mathfrak{a},M)$ is finitely generated for all $i\le m+n$. We define a class ${\text{cS}}_n(\mathfrak{a})$ of modules and we assume that $H_{\frak a}^s(M)\in\cS_{n}(\frak a)$forall$s\leq m$\mathrm{.}Weshowthat$H_{\frak a}^s(M)$is$\frak a$-cofiniteforall$s\leq m$ifeither$n=1$or$n\geq 2$and$\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$isfinitelygeneratedforall$1\leq t\leq n-1$,$i\leq t-1$and$s\leq m$\mathrm{.}If$A$isaringofdimension$d$and$M\in\cS_n(\frak a)$foranyideal$\frak a$ofdimension$\leq d-1$, then we prove that $M\in {\text{cS}}_n(\mathfrak{a})$ for any ideal $\mathfrak{a}$ of $A$.