Berkovich Motives
Peter Scholze
TL;DR
The paper constructs etale Berkovich motives using a Banach-ring, arc-topology framework that unifies archimedean and nonarchimedean geometry. It develops a robust six-functor formalism for motivic sheaves, proves a full cancellation theorem and rigidity of the motivic category over suitable bases, and relates Berkovich motives to classical étale motives via K-theory and nearby cycles. Central tools include finitary arc-sheaves, free motivic sheaves, and a Tate twist formalism that interact with algebraic K-theory to recover rational motivic cohomology. The work provides a self-contained analytic approach to motives that recovers Voevodsky’s theory over discrete fields and connects to tilting and perfectoid-type descent, with broad potential for applications to local Langlands and arithmetic geometry.
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.