| Title:
|
The universal tropicalization and the Berkovich analytification (English) |
| Author:
|
Giansiracusa, Jeffrey |
| Author:
|
Giansiracusa, Noah |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
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1805-949X (online) |
| Volume:
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58 |
| Issue:
|
5 |
| Year:
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2022 |
| Pages:
|
790-815 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
tropical geometry |
| Keyword:
|
tropical schemes |
| Keyword:
|
idempotent semirings |
| Keyword:
|
Berkovich analytification |
| Keyword:
|
semivaluation |
| MSC:
|
14G22 |
| MSC:
|
14T05 |
| idZBL:
|
Zbl 07655860 |
| idMR:
|
MR4538626 |
| DOI:
|
10.14736/kyb-2022-5-0790 |
| . |
| Date available:
|
2023-01-23T16:35:51Z |
| Last updated:
|
2023-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151304 |
| . |
| Reference:
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| . |
