Adic tropicalizations and cofinality of Gubler models

TL;DR

This paper introduces adic tropicalizations for subschemes of toric varieties, demonstrating their role as limits of Gubler models and establishing a natural isomorphism with Huber's adic analytification.

Contribution

It presents the concept of adic tropicalizations, proves cofinality of Gubler models for various schemes, and relates Berkovich analytifications to limits of tropicalizations.

Findings

Adic tropicalizations are limits of Gubler models.

Huber's adic analytification is isomorphic to the inverse limit of adic tropicalizations.

Berkovich analytifications are limits of tropicalizations in topologically ringed topoi.

Abstract

We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a G-topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi.