2008/12/17 by Roozbeh Hazrat, R. Hazrat, Adrian R. Wadsworth +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #math.RA
paper · pdf · doi:10.48550/arxiv.0812.3433
35 pages
arxiv created 2008/12/17 · arxiv updated 2009/12/01
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.