On the first Steenrod square for Chow groups

TL;DR

This paper constructs a weak homological Steenrod square for Chow groups without characteristic restrictions and applies it to generalize a theorem on quadratic forms' Witt index parity.

Contribution

It introduces a new weak Steenrod square for Chow groups applicable in all characteristics and extends Karpenko's theorem to characteristic two fields.

Findings

Constructed a weak homological Steenrod square for Chow groups.

Generalized Karpenko's theorem on Witt index parity to characteristic two.

No assumptions on the base field's characteristic are needed.

Abstract

We construct a weak version of the homological first Steenrod square, a natural transformation from the modulo two Chow group to the Chow group modulo two and two-torsion. No assumption is made on the characteristic of the base field. As an application, we generalize a theorem of Nikita Karpenko on the parity of the first Witt index of quadratic forms to the case of a base field of characteristic two.