$K$-Theory of Azumaya algebras

Abstract

For an Azumaya algebra $A$ which is free over its centre $R$, we prove that the $K$-theory of $A$ is isomorphic to $K$-theory of $R$ up to its rank torsion. We observe that a graded central simple algebra, graded by an abelian group, is a graded Azumaya algebra and it is free over its centre. So the above result, from the non-graded setting, covers graded central simple algebras. For a graded central simple algebra $A$, we can also consider graded projective modules. Let $\Pgr (R)$ be the category of graded finitely generated projective $R$-modules and $K_i, i\geq 0$, be the Quillen $K$-groups. Then $K_i^{\gr} (R)$ is defined to be $K_i( \Pgr (R))$. We give some examples to show that the graded $K$-theory of $A$ does not necessarily coincide with its usual $K$-theory. For a graded Azumaya algebra $A$, free over its centre $R$ and subject to some conditions, we show that $K_i^{\gr} (A)$ is ``very close'' to $K_i^{\gr}(R)$. Further, we consider additive commutators in the setting of graded division algebras. For a graded division algebra $D$ with a totally ordered abelian grade group, we show how the submodule generated by the additive commutators in $QD$ relates to that of $D$, where $QD$ is the quotient division ring.