Local homology, finiteness of Tor modules and cofiniteness

Abstract

Let [Formula: see text] be an ideal of a commutative noetherian ring [Formula: see text] with unity and [Formula: see text] an [Formula: see text]-module supported at [Formula: see text]. Let [Formula: see text] be the supermum of the integers [Formula: see text] for which [Formula: see text]. We show that [Formula: see text] is [Formula: see text]-cofinite if and only if the [Formula: see text]-module [Formula: see text] is finitely generated for every [Formula: see text]. This provides a hands-on and computable finitely-many-steps criterion to examine [Formula: see text]-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.