Berkovich Motives

TL;DR

This paper develops a unified theory of Berkovich motives that applies to both archimedean and nonarchimedean settings, extending Voevodsky's etale motives and ensuring key properties like cancellation and rigidity.

Contribution

It introduces a self-contained framework for Berkovich motives, bridging gaps between different analytic contexts and establishing foundational properties without relying on prior algebraic or analytic motive theories.

Findings

The cancellation theorem holds in this setting.

The stable ∞-category of motivic sheaves is rigid dualizable under mild conditions.

The theory recovers the etale version of Voevodsky's motives over discrete fields.

Abstract

We construct a theory of (etale) Berkovich motives. This is closely related to Ayoub's theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. Applying the theory to discrete fields, one still recovers the etale version of Voevodsky's theory. Two notable features of our setting which do not hold in other settings are that over any base, the cancellation theorem holds true, and under only minor assumptions on the base, the stable $\infty$-category of motivic sheaves is rigid dualizable.