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Variations on a theme of rationality of cycles

Nikita Karpenko — 2013

Open Mathematics

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 ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s...

Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic algebras.

Nikita A. Karpenko — 1995

Manuscripta mathematica

Holes in $I^{n}$

Nikita A. Karpenko — 2004

Annales scientifiques de l'École Normale Supérieure

Order of torsion in $C H^{4}$ of quadrics.

Karpenko, Nikita A. — 1996

Documenta Mathematica

On standard norm varieties

Nikita A. Karpenko; Alexander S. Merkurjev — 2013

Annales scientifiques de l'École Normale Supérieure

Let $p$ be a prime integer and $F$ a field of characteristic $0$ . Let $X$ be theof a symbol in the Galois cohomology group $H^{n + 1} (F, μ_{p}^{\otimes n})$ (for some $n \geq 1$ ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F (X)$ has the following property: for any equidimensional variety $Y$ , the change of field homomorphism $CH (Y) \to CH (Y_{F (X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (dim X) / (p - 1)$ . One of the main ingredients of the proof is a computation of...