Abstract
If M is a nonzero finitely generated module over a commutative Noetherian local ring R such that M has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that M has finite projective dimension, and hence a result of Foxby implies that R is Gorenstein. We prove that the same conclusion holds for certain nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general. Along the way, we also prove new results, independent of the reducing dimensions, concerning modules of finite Gorenstein dimension.
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Acknowledgements
The authors are grateful to Hiroki Matsui for pointing out several arguments used for the proof of Proposition 2.7. The authors also thank Henrik Holm, Arash Sadeghi, and Ryo Takahashi for helpful comments and discussions during the preparation of the manuscript.
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Toshinori Kobayashi was partly supported by JSPS Grant-in-Aid for JSPS Fellows 18J20660.
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Araya, T., Celikbas, O., Cook, J. et al. Modules with finite reducing Gorenstein dimension. Beitr Algebra Geom 65, 279–290 (2024). https://doi.org/10.1007/s13366-023-00687-x
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DOI: https://doi.org/10.1007/s13366-023-00687-x