Local homology, finiteness of Tor modules and cofiniteness

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\text{V}(\text{fa})$. Let $n$ be the supermum of the integers $i$ for which $H_i^{\text{fa}}(M)\mathrm{\ne }0$. We show that $M$ is $\text{fa}$-cofinite if and only if the $R$-module ${\text{Tor}}_i^R(R\mathrm{/}\text{fa},M)$ is finitely generated for every $0\le i\le n$. This provides a hands-on and computable finitely-many-steps criterion to examine $\mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.