Abstract
Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.
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Acknowledgements
Early versions of some of the results in this paper formed part of the second-named author’s PhD thesis at the University of Glasgow, funded by the EPSRC. The first author thanks Michael Wemyss and James Zhang for helpful conversations.
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Presented by Kenneth Goodearl.
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Brown, K.A., Macleod, M.J. The Cohen Macaulay Property for Noncommutative Rings. Algebr Represent Theor 20, 1433–1465 (2017). https://doi.org/10.1007/s10468-017-9694-z
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DOI: https://doi.org/10.1007/s10468-017-9694-z