Reduced K-theory of Azumaya Algebras, by Roozbeh Hazrat

In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are ``trivial'' in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps K_i(F) --> K_i(D), where K_i are Quillen K-groups, D is a division algebra and F its centre, or the homotopy fiber arising from the long exact sequence of above map, or reduced Whitehead group SK_1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is ``almost'' the same as K-theory of the base ring.

The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian local ring coincides with reduced K-theory of its residue central simple algebra.

The note ends with some calculation trying to determine the homotopy fibers mentioned above.



Roozbeh Hazrat <hazrat@maths.anu.edu.au>