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Gorenstein Injective Filtrations Over Cohen-Macaulay Rings with Dualizing Modules

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Abstract

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective R-modules need not be closed under tensor products, even when R is local and artinian; (2) the class of Gorenstein injective R-modules need not be closed under torsion products, even when R is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when R is a local, complete hypersurface.

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Acknowledgments

We are grateful to the anonymous referee for their helpful suggestions.

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Authors and Affiliations

  1. Physics Department, NDSU, Dept. 2755, P.O. Box 6050, Fargo, ND, 58108-6050, USA

    Aaron J. Feickert

  2. Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC, 29634, USA

    Sean Sather-Wagstaff

Authors
  1. Aaron J. Feickert
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  2. Sean Sather-Wagstaff
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Correspondence to Sean Sather-Wagstaff.

Additional information

Presented by Jon F. Carlson and Sean Sather-Wagstaff.

Dedicated to Edgar Enochs on the occasion of his retirement.

Sean Sather-Wagstaff was supported in part by NSA grant H98230-13-1-0215

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Feickert, A.J., Sather-Wagstaff, S. Gorenstein Injective Filtrations Over Cohen-Macaulay Rings with Dualizing Modules. Algebr Represent Theor 22, 297–319 (2019). https://doi.org/10.1007/s10468-018-9768-6

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  • DOI: https://doi.org/10.1007/s10468-018-9768-6

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