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

TL;DR

This paper proves the finiteness of the genus of finite-dimensional division algebras over finitely generated fields and explores applications to algebraic groups and K-theory, extending known results to more general fields.

Contribution

It establishes the finiteness of the genus for division algebras over finitely generated fields and discusses applications to algebraic groups and cohomological finiteness properties.

Findings

Finiteness of the genus for division algebras over finitely generated fields.

Connections between double cosets, cohomology, and finiteness properties.

Potential implications for algebraic group classification and K-theory.

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 $\textsf{G}_2$. 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 question over more general fields seem to come up for the first time. In the Appendix, we link the double cosets with $\check{\rm C}$ech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass's finiteness conjecture in $K$-theory.