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 work of V. Voevodsky.
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