Motives of Azumaya algebras

TL;DR

This paper investigates the slice filtration in K-theory and motives related to Azumaya algebras and Severi-Brauer varieties, applying conjectures to derive new results on algebraic K-theory invariants.

Contribution

It introduces a new analysis of the slice filtration for Azumaya algebras and motives, leading to the proof of vanishing results for SK_2 in certain cases.

Findings

Vanishing of SK_2(A) for central simple algebras with square-free index

Application of Beilinson-Lichtenbaum conjecture to motives

Analysis of the slice filtration for Azumaya algebras

Abstract

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK_2(A) for a central simple algebra A of square-free index.