Chow groups with twisted coefficients

TL;DR

This paper extends the concept of Chow groups with coefficients to non-torsion cases, computes these groups for classifying spaces of finite groups, and explores their relation to twisted motivic cohomology.

Contribution

It generalizes the definition of Chow groups with twisted coefficients and computes them for various finite groups, linking them to Serre's negligible cohomology and motivic cohomology.

Findings

Computed Chow groups for classifying spaces of finite groups.

Established a surjection from twisted motivic cohomology to Chow groups.

Connected twisted Chow groups to the theory of algebraic tori and coflasque resolutions.

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.