On the stable under specialization sets and cofiniteness of local cohomology

Abstract

Let R be a commutative Noetherian ring, be a stable under specialization subset of It is introduced in [15], the concept of -cofiniteness of an R-complex Also it is proved in [15, Theorem 4.7 (i) and (ii)] that if (a) R is semilocal with or (b) then the local cohomology module is -cofinite for all R-complexes and all In this paper, we continue to study -cofiniteness of an R-complex and we generalize (b) to a large class of complexes which recovers the condition of (b). We prove that is -cofinite for all whenever with We also prove that if is local, then for any complex is -cofinite.