Some Characterizations of Auslander and Bass Classes

Let R and S be rings and RCS a semidualizing bimodule. For a subcategory X of the Auslander class AC(S) containing all projective and C-injective modules, we show that a module N ∈ AC(S) if and only if there exists an exact sequence · · ·→ Xi → · · · → X1 → X0 → X0 → X1 → · · · → X^i → · · · in Mod S with all Xi , X^i in X such that it remains exact after applying the functor HomS(−, E) for any Cinjective module E and N ∼= Im(X0 → X^0). For a subcategory Y of the Bass class BC(R) containing all injective and C-projective modules, we show that a module M ∈ BC(R) if and only if there exists an exact sequence · · ·→Yi →· · ·→Y1 → Y0 → Y ^0 → Y ^1 → · · · → Y^i → · · · in Mod R with all Yi , Yi in Y such that it remains exact after applying the functor HomS(Q,−) for any C-projective module Q and M ∼= Im(Y0 → Y 0). We apply these results to comparison of some relative homological dimensions.