A criterion for cofiniteness of modules

Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $m,n$ non-negative integers. $M$ $A$-module such that $\operatorname{Ext}^i\_A(A/\mathfrak{a},M)$ is finitely generated for all $i\leq m+n$. We define class $\mathcal{S}n(\mathfrak{a})$ modules we assume $H{\mathfrak{a}}^s(M)\in\mathcal{S}{n}(\mathfrak{a})$ $s\leq m$. show $H{\mathfrak{a}}^s(M)$ $\mathfrak{a}$-cofinite m$ if either $n=1$ or $n\geq 2$ $\operatorname{Ext}A^{i}(A/\mathfrak{a},H{\mathfrak{a}}^{t+s-i}(M))$ $1\leq t\leq n-1$, t-1$ If ring dimension $d$ $M\in\mathcal{S}\_n(\mathfrak{a})$ any $\leq d-1$, then prove $A$.