Let R be a commutative noetherian ring, and let S (resp. L) be a Serre (resp. localizing) subcategory of the category of R-modules. If F is an unbounded complex of R-modules Tor-perpendicular to S and d is an integer, then i dSR F is in L for each R-module S in S if and only if i dk () R F is in L for each prime ideal such that R/ is in S, where k () is the residue field at. As an application, we show that for any R-module M, ₈ ₀R (k (), M) is in L for each prime ideal such that R/ is in S if and only if ^i 0R (S, M) is in L for each cyclic R-module S in S. We also obtain some new characterizations of regular and Gorenstein rings in the case of S consists of finite modules with supports in a specialization-closed subset V (I) of R.
Hashimoto et al. (Sat,) studied this question.