Totally Acyclic Approximations

Abstract

Let Q→R be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as a Q-bimodule admits a finite resolution by modules which are projective on both sides. We define an adjoint pair of functors between the homotopy category of totally acyclic R-complexes and that of Q-complexes. This adjoint pair is analogous to the classical adjoint pair of functors between the module categories of R and Q. As a consequence, we obtain a precise notion of approximations of totally acyclic R-complexes by totally acyclic Q-complexes