On the spinor norm and A0(X,K1) for quadrics

by Markus Rost (Preprint, September 88, 10 pages)

The K-Cohomology of generic splitting varieties for symbols in Milnor's K-theory plays an important role concerning the question of the bijectivity of the Galois symbol.

Let X be a smooth proper generic splitting variety for an n-fold symbol {a1,...,an} mod p. For n=2 one takes here Brauer-Severi varieties, for p=2 one takes Pfister quadrics; for general n and p it is not known how to construct such varieties.

A particular interesting case is then the group of "zero-cycles with coefficients in K1", denoted in the paper by A0(X,K1). This is the same thing as Hd(X,K{d+1}), with d=dim X. The major problem is here the injectivity of the direct image map N to K1 of the ground field.

The paper shows that N is injective for a certain range of quadrics, including the Pfister quadrics.

This result was basic for further computations of the K-Cohomology of Pfister quadrics by the author. It was also of use in the recent work of V. Voevodsky.

The considerations of the paper have been recently extendend by V. Chernousov and A. S. Merkuriev. Using the theory of exceptional Jordan algebras the author has proved the injectivity of N for the case n=p=3 (see the Tagungsbericht of the Oberwolfach conference "Algebraische K-Theorie", June 1996).

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