  Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and
$M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the
integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite
if and only if the $R$-module $\Tor^R_i(R/\fa,M)$ is finitely generated for
every $0\leq i\leq n$. 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.
