The Berkovich realization for rigid analytic motives

TL;DR

This paper demonstrates that the Berkovich realization functor is motivic and characterizes the maximal Artin quotient of a motive, extending Berkovich's results on weight-zero étale cohomology over non-archimedean fields.

Contribution

It introduces a motivic functor linking rigid analytic varieties to Berkovich topological spaces and generalizes key results on étale cohomology weight-zero parts.

Findings

The Berkovich realization functor is motivic.

It defines the maximal Artin quotient of a motive.

Generalizes Berkovich's results on weight-zero étale cohomology.

Abstract

We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the \'etale cohomology of a variety defined over a non-archimedean valued field.