Weil restriction and the motivic cycle class map
Qi Ge, Guangzhao Zhu
TL;DR
This work integrates motivic and étale viewpoints to study descent of cycle classes under finite Galois extensions. It first shows that the motivic cycle class map is induced by the étale cycle class map via the Geisser–Levine comparison, uniting motivic and Lichtenbaum cohomology through a natural comparison framework. The paper then constructs a Weil restriction on $\ell$-adic cohomology that mirrors Karpenko’s Weil restriction on algebraic cycles and proves its compatibility with the motivic cycle class map, providing a conceptual explanation for descent of cycle classes in this setting. Collectively, these results connect Chow groups, motivic cohomology, and $\ell$-adic cohomology under descent, clarifying descent phenomena in Beilinson–Lichtenbaum contexts and offering tools for understanding how cycle classes behave under finite Galois extensions.
Abstract
The etale cycle class map links Chow groups with etale cohomology. Within the framework of motivic cohomology, the motivic cycle class map appears as a comparison morphism relating motivic and Lichtenbaum cohomology. By examining the comparison morphism constructed by Geisser and Levine, we show that the motivic cycle class map is induced by the etale cycle class map. We then study the behavior of cycle class maps under Weil restriction. Extending Karpenko's construction of the Weil restriction for algebraic cycles and Chow groups, we introduce a corresponding Weil restriction for l-adic cohomology and prove its compatibility with the motivic cycle class map. This provides a conceptual explanation for the descent of cycle classes under finite Galois extensions.