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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 theintegers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofiniteif and only if the $R$-module $\Tor^R_i(R/\fa,M)$ is finitely generated forevery $0\leq i\leq n$. This provides a hands-on and computablefinitely-many-steps criterion to examine $\mathfrak{a}$-confiniteness. Ourapproach relies heavily on the theory of local homology which demonstrates theeffectiveness and indispensability of this tool.

Local homology, finiteness of Tor modules and cofiniteness