Acyclicity test of complexes modulo Serre subcategories using the residue fields

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 dS⊗R 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, i 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.