Reduced K-theory of Azumaya algebras

Abstract

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 center, or the homotopy fiber arising from the long exact sequence of above map, or the 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 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.

SK1 of Azumaya algebras over hensel pairs

“…The aim of this note is to prove that for the Hensel pair (R, I) where R is a semilocal ring, the map SK 1 (A) → SK 1 (A) is also an isomorphism. This extends a result of [6] to non-local Henselian rings.…”

On a theorem of Hazrat and Hoobler

“…They used the Skolem-Noether theorem. That case has also been proven by Hazrat [7] using the fact that A isétale locally a matrix algebra.…”

SK1 of Azumaya algebras over Hensel Pairs

“…Now consider the group CK 1 (A) = A * /R * A for the Azumaya algebra A over the Hensel pair (R, I ). A proof similar to Theorem 3.10 in [6], shows that CK 1 (A) ∼ = CK 1 (A). Thus in the case of tame unramified division algebra D, one can observe that CK 1 (D) ∼ = CK 1 (D).…”