Professor Jeffrey Giansiracusa jeffrey.giansiracusa@durham.ac.uk
Professor
Professor Jeffrey Giansiracusa jeffrey.giansiracusa@durham.ac.uk
Professor
Professor Jeffrey Giansiracusa jeffrey.giansiracusa@durham.ac.uk
Professor
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. 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 |
Public URL | https://durham-repository.worktribe.com/output/1194769 |
Published Journal Article
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