The universal tropicalization and the Berkovich analytification

Giansiracusa, Jeffrey; Giansiracusa, Noah

Abstract

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.

Citation

Giansiracusa, J., & Giansiracusa, N. (2022). The universal tropicalization and the Berkovich analytification. Kybernetika (Prague. On-line), 58(5), 790-815. https://doi.org/10.14736/kyb-2022-5-0790

• 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
• DOI: https://doi.org/10.14736/kyb-2022-5-0790