On rational equivalence in tropical geometry

TL;DR

This paper refines the concept of rational equivalence in tropical geometry, establishing foundational properties and computing Chow groups, with implications for the structure of tropical cycle intersection rings.

Contribution

It provides a corrected definition of rational equivalence, computes bounded Chow groups of 5^n, and proves that tropical cycles are sums of fan cycles, advancing the theoretical framework.

Findings

Bounded Chow groups of 5^n are isomorphic to fan cycles.

Every tropical cycle in 5^n is a sum of translated fan cycles.

The intersection ring of tropical cycles is generated in codimension 1.

Abstract

This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the "bounded" Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).