\[disc01xy\] Let $C$ be a semidualizing $R$-module.

\(a) The Auslander class ${{\mathcal{A}}_C}(R)$ contains all $R$-modules of finite projective dimension, and the Bass class ${{\mathcal{B}}_C}(R)$ contains all $R$-modules of finite injective dimension; see [@holm:fear Lemmas 4.1 and 5.1, and Corollary 6.3]. Moreover, it is straightforward to show that the Bass class satisfies the following local-global principal: For an $R$-module $M$, the following conditions are equivalent:

(i) $M\in{{\mathcal{B}}_C}(R)$;

(ii) $U^{-1}M\in{{\mathcal{B}}}_{U^{-1}C}(U^{-1}R)$ for each multiplicatively closed subset $U\subseteq R$;

(iii) $M_{{{\mathfrak{p}}}}\in{{\mathcal{B}}}_{C_{{{\mathfrak{p}}}}}(R_{{{\mathfrak{p}}}})$ for each ${{\mathfrak{p}}}\in{\operatorname{Spec}}(R)$; and

(iv) $M_{{{\mathfrak{m}}}}\in{{\mathcal{B}}}_{C_{{{\mathfrak{m}}}}}(R_{{{\mathfrak{m}}}})$ for each maximal ideal ${{\mathfrak{m}}}\in{\operatorname{Supp}}_R(M)$.

It follows that ${{\mathcal{B}}_C}(R)$ contains every $R$-module of locally finite injective dimension. The Auslander class satisfies an analogous local-global principal.

\(b) The Auslander and Bass classes also satisfy the two-of-three property by [@holm:fear Corollary 6.3]. That is, given a short exact sequence $0\to M'\to M\to M''\to 0$ of $R$-module homomorphisms, if two of the modules in the sequence are in ${{\mathcal{A}}_C}(R)$, then so is the third module, and similarly for ${{\mathcal{B}}_C}(R)$.

\(c) From [@takahashi:hasm (2.8)] we know that an $R$-module $M$ is in ${{\mathcal{B}}_C}(R)$ if and only if ${\operatorname{Hom}_{R}(C,M)}\in{{\mathcal{A}}_C}(R)$, and that $M\in{{\mathcal{A}}_C}(R)$ if and only if $C\otimes_RM\in{{\mathcal{B}}_C}(R)$. This is known as *Foxby equivalence*.

\[notation08bx\] Let $C$ be a semidualizing $R$-module. A finitely 