The universal tropicalization and the Berkovich analytification
Jeffrey Giansiracusa, Noah Giansiracusa · Kybernetika · 2023
Given an integral scheme X over a non-archimedean valued field k, we construct a universal closed embedding of X into a k-scheme equipped with a model over the field with one element F1 (a generalization of a toric variety).An embedding into such an ambient space determines a tropicalization of X by previous work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification X an .Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme Trop univ (X) whose T-points give the analytification and that 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.