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The group \(SK_{2}\) of a biquaternion algebra. (Le groupe \(SK_{2}\) d’une algèbre de biquaternions.) (French. Abridged English version) Zbl 1057.11022

Let \(SK_2\) be the kernel of the reduced norm for \(K_2\) [A. A. Suslin, K-theory 1, 5–29 (1987; Zbl 0635.12015)]. The paper is devoted to relating \(SK_2\) with Galois cohomology. Namely, the author sketches the proof of the following theorem: Let \(F\) be a field of characteristic zero, containing an algebraically closed subfield, \(D\) be the biquaternion algebra \(( \frac{a,b}{F}) \otimes ( \frac{c,d}{F})\). Then there is an exact sequence \[ \text{ker} N_{q'} \longrightarrow SK_2D \longrightarrow H^5(F, \mathbb{Z}/2) \longrightarrow H^5 (F(q), \mathbb{Z}/2), \] where \(q\) is the Albert quadratic form \(\langle a, b, - ab, - c, - d, cd \rangle\), \(q'\) a codimension-one subform of \(q\), and \(N_{q'}: H^3(X_{q'}, \mathcal{K}_5) \to K_2F\) is the usual norm map in \(K\)-cohomology [M. Rost, Doc. Math. 1, No. 16, 319–393 (1996; Zbl 0864.14002)].
The proof is divided into three steps. The first step contains computations by using the coniveau spectral sequence and a “weight” spectral sequence in étale motivic cohomology in order to relate \(K\)-cohomology groups with Galois cohomology groups. During the second step the author establishes an isomorphism between the projective quadric \(X_q\) of an Albert form and the generalized Severi-Brauer variety SB\((2, D)\) of the associated biquaternion algebra \(D\). Then using this isomorphism and Panin’s decomposition [I. A. Panin, K-theory 8, No. 6, 541–585 (1994; Zbl 0854.19002)] some levels of the topological filtration in terms of groups involving \(K\)-theory of the field \(F\) and the \(K\)-theory groups of the algebra \(D\) are computed.
In the last part it is shown that \(H^2(X_q, \mathcal{K}_4) \simeq K_2(X_q)^{2/3}\). This isomorphism and the computations of step 2 yield the exact sequence \[ SK_2D \longrightarrow H^5(F, \mathbb{Z}/2) \longrightarrow H^5(F(q), \mathbb{Z}/2). \] Finally, by using a codimension one subform \(q'\) of \(q\), the author proves that there is a surjective morphism \(\text{ker}N_{q'} \to \text{ker}(SK_2 \to H^5(F, \mathbb{Z}/2)\), and thereby the main exact sequence.

MSC:

11E70 \(K\)-theory of quadratic and Hermitian forms
19C99 Steinberg groups and \(K_2\)
19E20 Relations of \(K\)-theory with cohomology theories
12G05 Galois cohomology
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References:

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