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|Paper IPM / M / 15523||
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 HiZ(X) is
Z-cofinite for every homologically left-bounded R-complex X whose homology modules are finitely generated
and every i ∈ \mathbbZ.
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