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When does the canonical module of a module have finite injective dimension?

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Foxby (Math Scand 2:175–186, 1971–1972) showed that a Cohen-Macaulay module over a Gorenstein local ring has finite projective dimension if and only if its canonical module has finite injective dimension. In this paper, we establish the result given by Foxby in a general setting. As a byproduct, some criteria to detect the Cohen-Macaulay property of a ring are provided in terms of intrinsic properties of certain local cohomology modules. Also, as an application, we show that any Cohen-Macaulay module that has a canonical module with finite injective dimension satisfies the Auslander–Reiten conjecture.

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Acknowledgements

The authors would like to dedicate this paper in memory to Professor Wolmer Vasconcelos. He was a great supporter for the authors’ visit to Purdue University in 2013, when the authors began their studies on local cohomology theory, central theme of this work. Also, the authors are grateful to the anonymous referee for her/his careful reading of this manuscript and the many deep suggestions and corrections.

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Authors and Affiliations

  1. Universidade Tecnológica Federal do Paraná, Guarapuava, PR, 85053-525, Brazil

    T. H. Freitas

  2. Universidade de São Paulo - ICMC, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    V. H. Jorge Pérez

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  1. T. H. Freitas
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  2. V. H. Jorge Pérez
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Correspondence to T. H. Freitas.

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All authors were partially supported by CNPq-Brazil 421440/2016-3.

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Freitas, T.H., Jorge Pérez, V.H. When does the canonical module of a module have finite injective dimension?. Arch. Math. 117, 485–494 (2021). https://doi.org/10.1007/s00013-021-01659-0

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