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2 results

Motives of Azumaya algebras
Part of
- $K$-theory in geometry
- Cycles and subschemes
Bruno Kahn, Marc Levine
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 9 / Issue 3 / July 2010

Published online by Cambridge University Press:

14 April 2010, pp. 481-599

Print publication:

July 2010
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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.

Uniqueness of Moore's higher reciprocity law at the prime 2 for real number fields
Xuejun Guo, Hourong Qin
Journal:

Journal of K-Theory / Volume 1 / Issue 1 / February 2008

Published online by Cambridge University Press:

07 January 2008, pp. 185-192

Print publication:

February 2008
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Let F be a real number field with r1 real embeddings. In this paper, we prove that the sequence
is a complex, where are the Tate cohomology groups. Moreover if i ≡ 0, 1, or 2 (mod 4), then it is exact; if i ≡ 3 (mod 4), then the homology group at the second term of this complex is isomorphic to .