Vector bundles on tropical schemes

TL;DR

This paper develops a theory of vector bundles on tropical schemes, connecting algebraic, topological, and classical geometric perspectives, and shows how line bundles on tropical schemes relate to those on classical schemes.

Contribution

It introduces the concept of vector bundles on tropical schemes, relating them to existing theories and demonstrating line bundle lifting in the affine case.

Findings

Vector bundles on tropical schemes are defined and studied.

Line bundles on tropical schemes can be lifted to classical schemes in affine cases.

The algebraic background involves free modules over zero-sum free semirings.

Abstract

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.