A Note on K-Theory of Azumaya Algebras

Abstract

For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K _i (A, ℤ/m) = K _i (R, ℤ/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group.

Key Words:

• Azumaya algebras
• K-Theory

2000 Mathematics Subject Classification:

• Primary 16H05
• Secondary 19D99, 18F25

ACKNOWLEDGMENTS

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1. Part of this work has been done in the Winter of 2007 at KIAS, Seoul, Korea and the Summer of 2008 at ICTP, Trieste, Italy. The authors would like to thank Raymond Hoobler for several discussions regarding the subject.

Notes

Communicated by V. A. Artamonov.