On the first Steenrod square for Chow groups

Abstract

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.

Motivic Steenrod operations in characteristic $p$

“…In characteristic = 2, Hoffmann's conjecture was proved by Karpenko in [18]. Previously, Haution used a weak form of the first homological Steenrod square to prove a result on the parity of the first Witt index for nonsingular anisotropic quadratic forms over a field of characteristic 2 [14,Theorem 6.2]. Haution's result is a corollary of the case of Hoffmann's conjecture proved in this paper.…”

Motivic Steenrod operations in characteristicp

“…Using algebraic methods, Scully proved Conjecture 1.1 for totally singular anisotropic quadratic forms over base fields of characteristic 2 [28]. We also remark that Haution previously used a weak form of the first homological Steenrod square to prove a result on the parity of the first Witt index for nonsingular anisotropic quadratic forms over a field of characteristic 2 [14,Theorem 6.2].…”

A Bound for the Index of a Quadratic Form After Scalar Extension to the Function Field of a Quadric

“…Again, the index being 'close' to its maximal value is analogous to the situation in characteristic = 2 where one form becomes 'almost hyperbolic' over the function field of another quadric. 4 Example 6.24. We can illuminate Theorem 6.21 further by working through a concrete example, namely, that in which p is an (n + 1)-fold quasi-Pfister neighbour, i.e., dim( p) > 2 n and p is similar to a subform of an anisotropic (n + 1)-fold quasi-Pfister form (for example, p could itself be a quasi-Pfister form).…”

“…Here the notation v 2 (n) stands for the 2-adic order of the integer n. It is worth remarking that while Hoffmann's conjecture is essentially wide open for non-quasilinear forms in characteristic 2, non-trivial partial results have been obtained by Hoffmann-Laghribi [9] and also by Haution as a by-product of his efforts to develop the geometric machinery which is currently absent from the characteristic-2 setting (see [4,5]). Theorems 1.1 and 1.2 represent important landmarks for the theory of quadratic forms.…”