karpenko at math.jussieu.fr

Submission: 2011, Sep 21

We prove certain weak versions of some celebrated results due to Alexander Vishik
comparing rationality of algebraic cycles over the function field of a quadric
and over the base field. The original proofs use Vishik's symmetric operations
in the algebraic cobordism theory and work only in characteristic 0.
Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in
any characteristic different from 2. Our weak versions are still sufficient for
all existing applications. In particular, Vishik's construction of fields of
u-invariant 2^{r}+1 (for r>2) is extended to arbitrary characteristic not 2.

2010 Mathematics Subject Classification: 14C25; 11E04

Keywords and Phrases: Chow groups, quadrics, Steenrod operations, u-invariant.

Full text: dvi.gz 28 k, dvi 75 k, ps.gz 838 k, pdf.gz 153 k, pdf 189 k.

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