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Motives of Azumaya algebras

Part of: Cycles and subschemes $K$-theory in geometry

Published online by Cambridge University Press:  14 April 2010

Bruno Kahn  and
Marc Levine
Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu, 175–179 rue du Chevaleret, 75013 Paris, France ([email protected])
Marc Levine
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, USA ([email protected])
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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 SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Keywords

Bloch-Lichtenbaum spectral sequencealgebraic cyclesMorel–Voevodsky stable homotopy categoryslice filtrationAzumaya algebrasSeveri–Brauer schemes

MSC classification

Secondary: 14C25: Algebraic cycles 19E08: $K$-theory of schemes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 3 , July 2010 , pp. 481 - 599
Copyright
Copyright © Cambridge University Press 2010

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