Symmetric Auslander and Bass categories

Source abstract

Abstract We define the symmetric Auslander category A s ( R ) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and 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 s ( R ) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories where GMor ( R ) is the stable category of Gorenstein projective objects in the abelian category Mor ( R ) of homomorphisms of R -modules. This result is set in the wider context of a theory for A s ( R ) and B s ( R ), the symmetric Bass category which is defined dually.