Abstract
Let \((R, \mathfrak {m})\) be a commutative Noetherian local ring, I an ideal of R and let M be a non-zero I-cofinite R-module. In this paper we show that if M has finite injective dimension, then \(\dim R/I\leqslant \mathrm{inj\, dim}\, M \leqslant \textrm{depth}\, R\); and \(\mathrm{inj\, dim }\,M=\textrm{depth}\,R\), whenever \(\mathfrak {m} M \ne M\). These generalize the classical Bass formulas for injective dimension. As an application we obtain some results on the injective dimension of local cohomology modules. In addition, we show that R is a Cohen–Macaulay ring if admits a Cohen–Macaulay R-module of finite projective dimension.
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Acknowledgements
The authors are deeply grateful to the referee for his/her careful reading and helpful suggestions on the paper. We also would like to thank Prof. Hossein Zakeri for his reading of the first draft and valuable discussions.
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Asghari, F., Naghipour, R. & Sedghi, M. Injective dimension of cofinite modules and local cohomology. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 108 (2024). https://doi.org/10.1007/s13398-024-01610-2
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DOI: https://doi.org/10.1007/s13398-024-01610-2