Stable under specialization sets and cofiniteness

Abstract

Let [Formula: see text] be a commutative noetherian ring, and [Formula: see text] a stable under specialization subset of [Formula: see text]. We introduce a notion of [Formula: see text]-cofiniteness and study its main properties. In the case [Formula: see text], or [Formula: see text], or [Formula: see text] is semilocal with [Formula: see text], we show that the category of [Formula: see text]-cofinite [Formula: see text]-modules is abelian. Also, in each of these cases, we prove that the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for every homologically left-bounded [Formula: see text]-complex [Formula: see text] whose homology modules are finitely generated and every [Formula: see text].