Abstract
Let \(R\) be a left Noetherian ring satisfying the Auslander condition. It is proven that a left \(R\)-module \(M\) is Gorenstein injective if and only if \(M\) is strongly cotorsion and \(\mathrm{Ext}_R^{i\geq1}(I,M)=0\) for any injective left \(R\)-module \(I\); a right \(R\)-module \(M\) is Gorenstein flat if and only if \(M\) is strongly torsion-free and Tor\(^R_{i\geq1}(M,J)=0\) for any injective left \(R\)-module \(J\). We also prove that if \(R\) is a commutative Noetherian ring with splf \(R\) finite, then the local ring \(R_{\mathfrak{p}}\) is Gorenstein for every prime ideal \(\mathfrak{p}\) of \(R\) if and only if the cycles of every acyclic complex of PGF-modules are PGF-modules if and only if every complex of PGF-modules is a dg-PGF complex.
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Acknowledgments
The authors thank the referee for pertinent comments that improved the exposition.
Funding
This work was supported in part by NSF of China (grant no. 12261056).
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Wu, D. Auslander Condition and Gorenstein Rings. Math Notes 112, 789–796 (2022). https://doi.org/10.1134/S0001434622110141
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DOI: https://doi.org/10.1134/S0001434622110141