Tropical adic spaces I: The continuous spectrum of a topological semiring

TL;DR

This paper develops a tropical analogue of adic spaces by studying prime congruences on topological semirings, especially convergent power series, revealing meaningful geometric information in tropical toric cases.

Contribution

It introduces the theory of topological idempotent semirings and constructs a tropical adic space framework, extending classical valuation theory to tropical geometry.

Findings

Semiring of convergent power series can be endowed with a meaningful topology.

In tropical toric cases, the constructed spaces encode significant geometric data.

The dimension of these tropical spaces behaves as expected, aligning with classical intuition.

Abstract

Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow up paper.