Stable Under Specialization Sets and Cofiniteness

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