SK₂ of a biquaternion algebra

Journal für die Reine und Angewandte Mathematik, issue 605, 2007, p. 121-235
(1^st online appearance of the preprint on the 2-9-2003)

The main theorem of this article proves an exact sequence relating SK, kernel of the reduced norm for the K-theory of a biquaternion algebra, and a group of Galois cohomology. It can be stated as: Let

• F be a field of characteristic not 2 containing an algebraically closed subfield
• q be an Albert quadratic form (6 dimensional) and q' a codimension 1 subform
• D be the biquaternion algebra associated to q (essentially its Clifford invariant),
• F(q) be the function field of the projective quadric of zeros of q,

then, for i=0, 1 or 2, there is an exact sequence:

where the first group is a kernel in K-cohomology (kernel of the norm on the last term in the Gersten complex for the projective quadric associated to q').
This theorem generalizes to K₂ a theorem of M. Rost (see [1]) and its proof follows the same main steps. However, the group K₂ is much less understood than K₁, so the technical tools involved in the proofs are different from the ones used by M. Rost. For example, motivic cohomology is used, as well as algebraic group techniques such as the exceptionnal isomorphism between SL₄ and Spin₆.

[1] A. S. Merkurjev, K-theory of Simple Algebras, K-theory and Algebraic Geometry: connections with quadratic forms and division algebras, Proc. Symp. Pure Math., vol. 1, n. 58 (1995), p. 65-83