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The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

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Abstract

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link thedoublecosets with Čech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass’s finiteness conjecture in K-theory.

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Acknowledgements

The first author was supported by an NSERC research grant. During the preparation of the paper, the second author visited Princeton University and the Institute for Advanced Study on a Simons Fellowship; the hospitality of both institutions and the generous support of the Simons Foundation are thankfully acknowledged. The third author was partially supported by an AMS-Simons Travel Grant. We would like to thank the anonymous referee for offering a number of corrections and valuable suggestions. We are also grateful to Louis Rowen for useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of Alberta Edmonton, Edmonton, Alberta, T6G 2G1, Canada

    Vladimir I. Chernousov

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904-4137, USA

    Andrei S. Rapinchuk

  3. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Igor A. Rapinchuk

Authors
  1. Vladimir I. Chernousov
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  2. Andrei S. Rapinchuk
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  3. Igor A. Rapinchuk
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Correspondence to Andrei S. Rapinchuk.

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To Louis Rowen on the occasion of his retirement

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Chernousov, V.I., Rapinchuk, A.S. & Rapinchuk, I.A. The finiteness of the genus of a finite-dimensional division algebra, and some generalizations. Isr. J. Math. 236, 747–799 (2020). https://doi.org/10.1007/s11856-020-1988-x

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  • DOI: https://doi.org/10.1007/s11856-020-1988-x

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