On Graded Simple Algebras

TL;DR

This paper explores properties of graded simple algebras, showing that graded central simple algebras are graded Azumaya algebras and analyzing their K-theory, revealing their close relationship with the K-theory of their centers.

Contribution

It demonstrates that graded central simple algebras are graded Azumaya and establishes a connection between their K-theory and that of their centers.

Findings

Graded central simple algebras are graded Azumaya.

K_i^{ ext{gr}}(A) is closely related to K_i^{ ext{gr}}(R).

The K-theory of graded Azumaya algebras approximates that of their centers.

Abstract

This note begins by observing that a graded central simple algebra, graded by an abelian group, is a graded Azumaya algebra and it is free over its centre. For a graded Azumaya algebra A free over its centre $R$, we show that K_i^{\gr} (A) is "very close" to K_i^{\gr}(R), where K_i^{\gr} (R) is defined to be K_i(\Pgr (R)). Here \Pgr (R) is the category of graded finitely generated projective R-modules and K_i, \,i\geq 0, are the Quillen K-groups.