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On the first Steenrod square for Chow groups

Olivier Haution

TL;DR

This paper constructs a weak first homological Steenrod square on modulo two Chow groups that is well-defined over fields of arbitrary characteristic. The construction relies on a 2-integrality property of the homological Chern character and yields a natural transformation ${\mathrm Sq}_1: {\mathrm Ch}_\bullet\to {\widetilde{\mathrm Ch}}_{\bullet-1}$, with compatibility under proper push-forwards, base change, and external products; for smooth varieties it satisfies ${\mathrm Sq}_1^X[X]=\widetilde{c_1(T_X)}$. The work enables a characteristic-free analysis of quadratic forms, culminating in a parity result for the first Witt index that parallels Hoffmann’s conjecture in non-characteristic two, thereby contributing to a uniform framework for quadratic forms across characteristics. The methods blend an algebraic Riemann-Roch perspective with Chow-theoretic and K-theoretic filtrations to control torsion phenomena and to relate cycle-theoretic operations to tangent-canonical data.

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.

On the first Steenrod square for Chow groups

TL;DR

This paper constructs a weak first homological Steenrod square on modulo two Chow groups that is well-defined over fields of arbitrary characteristic. The construction relies on a 2-integrality property of the homological Chern character and yields a natural transformation , with compatibility under proper push-forwards, base change, and external products; for smooth varieties it satisfies . The work enables a characteristic-free analysis of quadratic forms, culminating in a parity result for the first Witt index that parallels Hoffmann’s conjecture in non-characteristic two, thereby contributing to a uniform framework for quadratic forms across characteristics. The methods blend an algebraic Riemann-Roch perspective with Chow-theoretic and K-theoretic filtrations to control torsion phenomena and to relate cycle-theoretic operations to tangent-canonical data.

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.
Paper Structure (11 sections, 10 theorems, 42 equations)

This paper contains 11 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Grothendieck group of coherent sheaves
  4. Topological filtration
  5. External product
  6. Comparison with Chow groups
  7. Scalars extension
  8. Singular Grothendieck-Riemann-Roch Theorem
  9. A $2$-integrality property of the homological Chern character
  10. The first homological Steenrod square
  11. Parity of the first Witt index of a quadratic form

Key Result

Theorem 3.1

For all varieties $X$ there is a homomorphism with the following properties.

Theorems & Definitions (22)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 12 more