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Chow groups with twisted coefficients

Burt Totaro

TL;DR

This work extends Rost's Chow groups with twisted coefficients to arbitrary locally constant étale sheaves $E$, including non-torsion, and develops a robust toolkit (coflasque resolutions, transfers, and equivariant theories) to compute these groups and connect them to negligible cohomology and covering-space descriptions. It then introduces twisted motivic cohomology and shows a natural surjection to twisted Chow groups, but provides concrete counterexamples where the map fails to be an isomorphism, clarifying the landscape between motivic and étale perspectives. The paper also establishes an étale cycle map, purity results, and a cycle map for complex varieties, and analyzes the behavior of twisted Chow groups in key families (cyclic, generalized quaternion, and Klein groups), including exact generators, bounds, and codimension-1 injectivity versus higher-codimension failures. Overall, twisted Chow groups are shown to interpolate between twisted motivic cohomology and twisted étale cohomology, offering concrete computational tools and revealing nuanced interactions among transfers, coflasque resolutions, and covering-space descriptions.

Abstract

Rost defined the Chow group of algebraic cycles with coefficients in a locally constant torsion etale sheaf. We generalize the definition to allow non-torsion coefficients. Chow groups with twisted coefficients are related to Serre's notion of "negligible cohomology" for finite groups. We generalize a computation by Merkurjev and Scavia of negligible cohomology, in terms of twisted Chow groups. We compute the Chow groups of the classifying space BG with coefficients in an arbitrary G-module, for several finite groups G (cyclic, quaternion, ${\bf Z}/2\times {\bf Z}/2$). There are connections with the theory of algebraic tori, notably the concept of coflasque resolutions. We compare twisted Chow groups with twisted motivic cohomology as defined by Heller-Voineagu-Ostvaer. Surprisingly, there is a surjection from twisted motivic cohomology to twisted Chow groups, but it is not always an isomorphism.

Chow groups with twisted coefficients

TL;DR

This work extends Rost's Chow groups with twisted coefficients to arbitrary locally constant étale sheaves , including non-torsion, and develops a robust toolkit (coflasque resolutions, transfers, and equivariant theories) to compute these groups and connect them to negligible cohomology and covering-space descriptions. It then introduces twisted motivic cohomology and shows a natural surjection to twisted Chow groups, but provides concrete counterexamples where the map fails to be an isomorphism, clarifying the landscape between motivic and étale perspectives. The paper also establishes an étale cycle map, purity results, and a cycle map for complex varieties, and analyzes the behavior of twisted Chow groups in key families (cyclic, generalized quaternion, and Klein groups), including exact generators, bounds, and codimension-1 injectivity versus higher-codimension failures. Overall, twisted Chow groups are shown to interpolate between twisted motivic cohomology and twisted étale cohomology, offering concrete computational tools and revealing nuanced interactions among transfers, coflasque resolutions, and covering-space descriptions.

Abstract

Rost defined the Chow group of algebraic cycles with coefficients in a locally constant torsion etale sheaf. We generalize the definition to allow non-torsion coefficients. Chow groups with twisted coefficients are related to Serre's notion of "negligible cohomology" for finite groups. We generalize a computation by Merkurjev and Scavia of negligible cohomology, in terms of twisted Chow groups. We compute the Chow groups of the classifying space BG with coefficients in an arbitrary G-module, for several finite groups G (cyclic, quaternion, ). There are connections with the theory of algebraic tori, notably the concept of coflasque resolutions. We compare twisted Chow groups with twisted motivic cohomology as defined by Heller-Voineagu-Ostvaer. Surprisingly, there is a surjection from twisted motivic cohomology to twisted Chow groups, but it is not always an isomorphism.

Paper Structure

This paper contains 17 sections, 28 theorems, 112 equations.

Table of Contents

  1. Definition
  2. Basic calculations
  3. Chow groups and coflasque resolutions
  4. Twisted motivic cohomology
  5. Equivariant Chow groups with coefficients
  6. The étale cycle map
  7. The cycle map for complex varieties
  8. Twisted Chow groups and transfers
  9. Twisted Chow groups of a cyclic group
  10. Twisted Chow groups in codimension 1
  11. The generalized quaternion groups
  12. Bounds on equivariant Chow groups with coefficients
  13. The group $\text{\bf Z}/2\times \text{\bf Z}/2$
  14. Twisted Chow groups vs. twisted motivic cohomology
  15. The residue on étale motivic cohomology
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $f\colon Y\to X$ be a finite étale morphism of schemes of finite type over a field $k$. For a locally constant étale sheaf $E$ on $Y$,

Theorems & Definitions (69)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 59 more