Some Characterizations of Auslander and Bass Classes

Abstract

Let R and S be rings and $_RC_S$ a semidualizing bimodule. For a subcategory ${\mathcal {X}}$ of the Auslander class ${\mathcal {A}}_C(S)$ containing all projective and C-injective modules, we show that a module $N\in {\mathcal {A}}_C(S)$ if and only if there exists an exact sequence $\cdots \rightarrow X_i\rightarrow \cdots \rightarrow X_1\rightarrow X_0\rightarrow X^0\rightarrow X^1\rightarrow \cdots \rightarrow X^i\rightarrow \cdots$ in $\mathrm{Mod}\,S$ with all $X_i,X^i$ in ${\mathcal {X}}$ such that it remains exact after applying the functor $\mathrm{Hom}_S(-,E)$ for any C-injective module E and $N\cong \mathrm{Im}(X_0\rightarrow X^0)$. For a subcategory ${\mathcal {Y}}$ of the Bass class ${\mathcal {B}}_C(R)$ containing all injective and C-projective modules, we show that a module $M\in {\mathcal {B}}_C(R)$ if and only if there exists an exact sequence $\cdots \rightarrow Y_i\rightarrow \cdots \rightarrow Y_1\rightarrow Y_0\rightarrow Y^0\rightarrow Y^1\rightarrow \cdots \rightarrow Y^i\rightarrow \cdots$ in $\mathrm{Mod}\,R$ with all $Y_i,Y^i$ in ${\mathcal {Y}}$ such that it remains exact after applying the functor $\mathrm{Hom}_S(Q,-)$ for any C-projective module Q and $M\cong \mathrm{Im}(Y_0\rightarrow Y^0)$. We apply these results to comparison of some relative homological dimensions.