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.

令R和S为环，\（_ RC_S \）为半对偶双模。对于包含所有射影和C射影模块的Auslander类\（{\ mathcal {A}} _ C（S）\）的子类别\ {{mathcal {X}} \ }，我们显示了一个模块\（N \ in {\ mathcal {A}} _ C（S）\）仅当存在精确序列\（\ 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 \）在\（\ mathrm {国防部} \，S \）与所有\（X_I，X ^ I \）在\（{\ mathcal {X}} \）这样，对于任何C内射模E和\（N \ cong \ mathrm {Im}（X_0 \ rightarrow X ^ 0），应用函子\（\ mathrm {Hom} _S（-，E）\）后，它仍然保持精确\）。对于包含所有射影和C射影模块的Bass类\（{\ mathcal {B}} _ C（R）\）的子类别\ {{\ mathcal {Y}} \ }，我们表明模块\ {M \ in {\ mathcal {B}} _ C（R）\）仅当存在精确序列\（\ cdots \ rightarrow Y_i \ rightarrow \ cdots \ rightarrow Y_1 \ rightarrow Y_0 \ rightarrow Y ^ 0 \ rightarrow Y ^ 1 \ RIGHTARROW \ cdots \ RIGHTARROW Y 1 I \ RIGHTARROW \ cdots \）在\（\ mathrm {国防部} \，R \）与所有\（Y_I，Y ^ I \）在\（{\ mathcal {Y}} \） ，使得其将所述后算符确切保持\（\ mathrm {坎} _S（Q， - ）\）对于任何Ç -projective模块Q和\（M \ cong \ mathrm {Im}（Y_0 \ rightarrow Y ^ 0）\）。我们将这些结果用于一些相对同源性尺寸的比较。