A noncommutative generalization of Auslander's last theorem

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finitely generated or not) were defined. In the same paper, the dual notion of a Gorenstein projective module was defined and so a relative theory of Gorenstein modules was initiated (cf. [2, 5] and references therein). In [12], Grothendieck introduced the notion of a dualizing complex. A dualizing module for R is one whose deleted injective resolution is a dualizing complex. Then a local Noetherian ring R is Gorenstein if and only if R is itself a dualizing module for R. In this case, Auslander announced the result that over such a ring, every finitely generated module has a finitely generated Gorenstein projective cover (or equivalently, a minimal maximal Cohen-Macaulay approximation). In [9], this result was generalized to the situation where R is a local Cohen-Macaulay ring having a dualizing module. More recently, in [13], Jørgensen has shown the existence of Gorenstein projective precovers for every module over a commutative Noetherian ring with a dualizing complex. Using Christensen [3], we here introduce the notion of a dualizing bimodule associated with a pair of Noetherian rings (but not necessarily commutative ones). In [6], it was shown that in this situation, every module in the Auslander class defined by the pair of rings admits a Gorenstein projective precover. Now we give examples where the dualizing bimodule has a double structure over the same noncommutative Noetherian ring and that in this case, if the ring also admits a Matlis dualizing module, (cf. [8] or [10]), we particularize the result to the existence of a stronger approximation, that is, every finitely generated module in the Auslander class has a finitely generated Gorenstein projective cover. Given a class of R-modules , an -precover of a left R-module M is a morphism