Weil restriction and the motivic cycle class map

TL;DR

This paper develops a framework connecting Weil restriction, motivic cohomology, and realization functors within the categories of motives, enhancing understanding of their intrinsic relationships in algebraic geometry.

Contribution

It constructs the Weil restriction map for l-adic and mixed Weil cohomology theories and interprets it within the triangulated categories of motives using Grothendieck's six-functor formalism.

Findings

Weil restriction map is compatible with the motivic cycle class map.

The construction admits a natural interpretation in the categories of motives.

The approach provides a conceptual understanding of the interaction between Weil restriction and motivic cohomology.

Abstract

We construct the Weil restriction map for l-adic cohomology and, more generally, for mixed Weil cohomology theories. We study its compatibility with the motivic cycle class map and show that these constructions admit a natural interpretation in the triangulated categories of motives. Using Grothendieck's six-functor formalism, we prove that the Weil restriction map arises intrinsically from the functorial structures of these categories. This provides a conceptual framework for understanding the interaction between Weil restriction, motivic cohomology, and realization functors.