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Keywords:
tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation
tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation
Summary:
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 $\mathbb{F}_1$ (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 $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme $\mathit{Trop}_{univ}(X)$ whose $\mathbb{T}$-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 $X=\mathrm{Spec}\: A$ is affine, we show that $\mathit{Trop}_{univ}(X)$ 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 $\mathit{Trop}_{univ}(X)$ represents the moduli functor of semivaluations on $X$, and when $X=\mathrm{Spec}\: A$ is affine there is a universal semivaluation on $A$ taking values in the idempotent semiring of regular functions on the universal tropicalization.
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