SK1 for graded division algebras

TL;DR

This paper investigates the reduced Whitehead group SK1 of graded division algebras, revealing simpler computations in the graded setting and establishing key isomorphisms with ungraded cases, including stability and formulas for valued division algebras.

Contribution

It proves that SK1 of a tame valued division algebra equals SK1 of its associated graded algebra and shows SK1 of a graded algebra is isomorphic to that of its quotient division algebra.

Findings

SK1 computations are more straightforward in the graded setting

SK1 of a tame valued division algebra matches that of its graded counterpart

SK1 of a graded algebra is isomorphic to SK1 of its quotient division algebra

Abstract

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.