2011/01/07 by Millar, Judith R · 1 citation
#FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1101.1468
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 Ki, i≥ 0, be the Quillen K-groups. Then Ki\gr (R) is defined to be Ki( \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 Ki\gr (A) is ``very close'' to Ki\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.