The universal tropicalization and the Berkovich analytification

TL;DR

This paper introduces a universal tropicalization framework that unifies Berkovich analytification with scheme-theoretic tropicalizations, providing a canonical and functorial perspective on non-archimedean geometry.

Contribution

It constructs a universal tropicalization scheme that captures the Berkovich analytification and relates all other tropicalizations, enriching the understanding of non-archimedean spaces.

Findings

Berkovich analytification equals the set-theoretic tropicalization of the universal embedding.

The universal tropicalization is the limit of all affine tropicalizations.

The universal tropicalization represents the moduli functor of semivaluations.

Abstract

Given an integral scheme X over a non-archimedean valued field k , we construct acuniversal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by earlier work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification of X. Moreover, using the scheme-theoretic tropicalization, we obtain a tropical scheme $Trop_{univ}(X)$ whose T-points give the analytification and which canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X = spec A is affine, we show that $Trop_{univ}(X)$ is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $Trop_{univ}(X)$ represents the moduli functor of semivaluations on X, and when X = spec A is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.