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,...,a_n} 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 H^d(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 work of V. Voevodsky.

Related:

• Using the theory of exceptional Jordan algebras the author has proved the injectivity of N for the case n=p=3. See my talk in the Tagungsbericht of the Oberwolfach conference "Algebraische K-Theorie", June 1996.
• Chernousov, Vladimir; Merkurjev, Alexander. R-equivalence in spinor groups. J. Amer. Math. Soc. 14 (2001), no. 3, 509-534. MR 1824991.

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