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Frobenius bimodules and flat-dominant dimensions

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Abstract

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Furthermore, let A and B be finite-dimensional algebras over a field k, and let domdim(AX) stand for the dominant dimension of an A-module X. If BMA is a Frobenius bimodule, then domdim(A) ⩽ domdim(BM) and domdim(B) ⩽ domdim(AHomB(M, B)). In particular, if B ⊆ A is a left-split (or right-split) Frobenius extension, then domdim(A) = domdim(B). These results are applied to calculate flat-dominant dimensions of a number of algebras: shew group algebras, stably equivalent algebras, trivial extensions and Markov extensions. We also prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

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Acknowledgements

This work was supported by the Beijing Natural Science Foundation (Grant No. 1192004). The author is grateful to the Alexander von Humboldt Foundation for supporting an academic visit to the University of Stuttgart in the summer of 2018, where part of the work was revised, and to Steffen Koenig for his friendship and hospitality.

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  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Changchang Xi

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Xi, C. Frobenius bimodules and flat-dominant dimensions. Sci. China Math. 64, 33–44 (2021). https://doi.org/10.1007/s11425-018-9519-2

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