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The Finiteness of the Genus of a Finite-Dimensional Division Algebra, and Some Generalizations
Israel Journal of Mathematics ( IF 1 ) Pub Date : 2020-03-01 , DOI: 10.1007/s11856-020-1988-x
Vladimir I. Chernousov , Andrei S. Rapinchuk , Igor A. Rapinchuk

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 G 2 . 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.

中文翻译:

有限维除法代数的属的有限性和一些推广

我们证明,当中心是任何特征的有限生成域时,有限维除法代数的属是有限的。我们还讨论了我们的方法在其他问题上的潜在应用,包括类型 G 2 的简单代数群的属的有限性。这些应用涉及任意有限生成域上代数群的 adele 群的双陪集:虽然在数字域上,这些双陪集与代数群的类数相关联,因此已被积极分析,但似乎出现了关于更一般领域的类似问题第一次起来。在附录中,我们将双陪集与 Čech 上同调联系起来,并指出了涉及双陪集(条件 (T))的某些有限性属性与 K 理论中的巴斯有限性猜想之间的联系。
更新日期:2020-03-01
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