Symmetric Auslander and Bass Categories

We define the symmetric Auslander category A(R) to consist of complexes of projective modules whose leftand righttails are equal to the leftand right-tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that A(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories GMor(R) ' → A(R)/K(Prj R) where GMor(R) is the stable category of Gorenstein projective objects in the abelian category Mor(R) of homomorphisms of Rmodules. This result is set in the wider context of a theory for A(R) and B(R), the symmetric Bass category which is defined dually.