Stable under specialization sets and cofiniteness.

Let R be a commutative noetherian ring, and 𝒵 a stable under specialization subset of Spec (R). We introduce a notion of 𝒵 -cofiniteness and study its main properties. In the case dim (𝒵) ≤ 1 , or dim (R) ≤ 2 , or R is semilocal with cd (𝒵 , R) ≤ 1 , we show that the category of 𝒵 -cofinite R -modules is abelian. Also, in each of these cases, we prove that the local cohomology module H 𝒵 i (X) is 𝒵 -cofinite for every homologically left-bounded R -complex X whose homology modules are finitely generated and every i ∈ ℤ. [ABSTRACT FROM AUTHOR]