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The universal tropicalization and the Berkovich analytification

Giansiracusa, Jeffrey; Giansiracusa, Noah

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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 [GG16], 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 of [GG16], we obtain a tropical scheme Tropuniv(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 = SpecA 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 = SpecA is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.


Giansiracusa, J., & Giansiracusa, N. (2022). The universal tropicalization and the Berkovich analytification. Kybernetika (Prague. On-line), 58(5), 790-815.

Journal Article Type Article
Acceptance Date Aug 8, 2022
Publication Date 2022
Deposit Date Aug 9, 2022
Publicly Available Date Mar 31, 2023
Journal Kybernetika
Print ISSN 0023-5954
Electronic ISSN 1805-949X
Publisher Institute of Information Theory and Automation, The Czech Academy of Sciences
Peer Reviewed Peer Reviewed
Volume 58
Issue 5
Pages 790-815


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