Injectivity of K₂(D) ⟶ K₂(F) for quaternion algebras

by Markus Rost (Preprint, May 86, 11 pages)

It is shown that the reduced norm K₂(D) → K₂(F) for a quaternion algebra D is injective. The proof uses Swan's computation of the K-theory of quadrics. There is not much difference to the proof given by A. S. Merkurjev in May 1986 which appeared in the articles

• Merkurjev, A. S., The group SK₂ for quaternion algebras. Dokl. Akad. Nauk SSSR 298 (1988), no. 3, 548-550; translation in Soviet Math. Dokl. 37 (1988), no. 1, 118-120. MR 925958.
• Merkurjev, A. S., The group SK₂ for quaternion algebras. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 2, 310-335. MR 941679.

An "elementary" proof of the result can be found in the preprint On Hilbert Satz 90 for K₃ for quadratic extensions.

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