The universal tropicalization and the Berkovich analytification

Jeffrey Giansiracusa; Noah Giansiracusa

Kybernetika (2022)

  • Volume: 58, Issue: 5, page 790-815
  • ISSN: 0023-5954

Abstract

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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 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 an . Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme 𝑇𝑟𝑜𝑝 u n i v ( X ) whose 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 = Spec A is affine, we show that 𝑇𝑟𝑜𝑝 u n i v ( 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 𝑇𝑟𝑜𝑝 u n i v ( X ) represents the moduli functor of semivaluations on X , and when X = Spec A is affine there is a universal semivaluation on A taking values in the idempotent semiring of regular functions on the universal tropicalization.

How to cite

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Giansiracusa, Jeffrey, and Giansiracusa, Noah. "The universal tropicalization and the Berkovich analytification." Kybernetika 58.5 (2022): 790-815. <http://eudml.org/doc/299458>.

@article{Giansiracusa2022,
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 $\mathbb \{F\}_1$ (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.},
author = {Giansiracusa, Jeffrey, Giansiracusa, Noah},
journal = {Kybernetika},
keywords = {tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation},
language = {eng},
number = {5},
pages = {790-815},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The universal tropicalization and the Berkovich analytification},
url = {http://eudml.org/doc/299458},
volume = {58},
year = {2022},
}

TY - JOUR
AU - Giansiracusa, Jeffrey
AU - Giansiracusa, Noah
TI - The universal tropicalization and the Berkovich analytification
JO - Kybernetika
PY - 2022
PB - Institute of Information Theory and Automation AS CR
VL - 58
IS - 5
SP - 790
EP - 815
AB - 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.
LA - eng
KW - tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation
UR - http://eudml.org/doc/299458
ER -

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