Reduced Steenrod operations and resolution of singularities

TL;DR

The paper introduces a new construction of reduced Steenrod operations for Chow groups modulo a prime p, applicable to certain varieties, with implications for quadratic forms.

Contribution

It provides a novel weak form of Steenrod operations for Chow groups, extending their applicability to varieties over fields with resolution of singularities.

Findings

Constructed reduced Steenrod operations for specific varieties.

Applied these operations to problems in quadratic form theory.

Extended the scope of Steenrod operations to new algebraic contexts.

Abstract

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.