Tropical analogs of Milnor $K$-groups and tropicalizations of Zariski-Riemann spaces

TL;DR

This paper develops tropical analogs of Milnor K-groups and establishes their Gersten resolution, aiming to advance tropical methods in algebraic cohomology and contribute to the tropical Hodge conjecture.

Contribution

It introduces tropical Milnor K-groups and proves their Gersten resolution, providing foundational tools for tropical approaches to algebraic cohomology problems.

Findings

Existence of Gersten resolution for tropical Milnor K-groups

Construction of tropical analogs of rational Milnor K-groups

Application to tropical Hodge conjecture in future work

Abstract

As a step of a tropical approach to problems on algebraic classes of cohomology groups (such as the Hodge conjecture), in this paper, we introduce tropical analogs of (rational) Milnor $K$-groups, and prove the existence of the Gersten resolution of their Zariski sheafification. Our result will be used to prove a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields in a subsequent paper.