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.

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Acknowledgements

The authors thank the referee for useful suggestions.

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Correspondence to Weiling Song.

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Communicated by Ergun Yalcin.

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Huang, Y., Song, W. Some Characterizations of Auslander and Bass Classes. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00517-z

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Keywords

  • Semidualizing bimodules
  • Auslander classes
  • Bass classes
  • Relative homological dimensions

Mathematics Subject Classification

  • 18G25
  • 16E10