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arXiv · 1802.09092

Noncommutative quasi-resolutions

X. -S. Qin·Y. -H. Wang·J. J. Zhang
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Abstract

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then anoncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given non-commutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a given non-commutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.

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X. -S. Qin, Y. -H. Wang, J. J. Zhang. 2018-02-25. Noncommutative quasi-resolutions. https://arxiv.org/abs/1802.09092

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