Chow groups with twisted coefficients

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.