THE UNIVERSAL TROPICALIZATION AND THE BERKOVICH ANALYTIFICATION

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 Xan. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme Tropuniv(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 Tropuniv(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 Tropuniv(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.