Local Tropicalization

TL;DR

This paper introduces a functorial framework for local tropicalization in commutative algebra, enabling the study of valuations and subvarieties in toroidal embeddings through polyhedral fans.

Contribution

It provides a general, functorial definition of local tropicalization that applies to valuations on rings and relates global and local tropicalizations in toric geometry.

Findings

Local tropicalizations are supports of finite rational polyhedral fans.

The framework generalizes tropicalization to local and toroidal settings.

Comparison between global and local tropicalizations is established.

Abstract

In this paper we propose a general functorial definition of the operation of \emph{local tropicalization} in commutative algebra. Let $R$ be a commutative ring, $\Gamma$ a finitely generated subsemigroup of a lattice, $\gamma : \Gamma \rightarrow R/ R^*$ a morphism of semigroups, and $\V(R)$ the topological space of valuations on $R$ taking values in $\R \cup \infty$. Then we may \emph{tropicalize} with respect to $\gamma$ any subset $\W$ of the space of valuations $\V(R)$. By definition, we get a subset of a rational polyhedral cone canonically associated to $\Gamma$, enriched with strata at infinity. In particular, when $R$ is a local ring, $\gamma$ is a \emph{local} morphism of semigroups, and $\W$ is the space of valuations which are either positive or non-negative on $R$, we call these processes \emph{local tropicalizations}. They depend only on the ambient toroidal structure, which in turn allows to define tropicalizations of subvarieties of toroidal embeddings. We prove that with suitable hypothesis, these local tropicalizations are the supports of finite rational polyhedral fans enriched with strata at infinity and we compare the global and local tropicalizations of a subvariety of a toric variety.