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Let  R  be a commutative Noetherian ring,be a stable under specialization subset ofIt is introduced in [ 15 ], the concept of-cofiniteness of an  R -complexAlso it is proved in [ 15 , Theorem 4.7 (i) and (ii)] that if (a)  R  is semilocal withor (b)then the local cohomology moduleis-cofinite for all  R -complexesand allIn this paper, we continue to study-cofiniteness of an  R -complexand we generalize (b) to a large class of complexes which recovers the condition of (b). We prove thatis-cofinite for allwheneverwithWe also prove that ifis local, then for any complexis-cofinite.

On the stable under specialization sets and cofiniteness of local cohomology