The Orlik-Solomon Algebra and the Bergman Fan of a Matroid

TL;DR

This paper establishes a canonical isomorphism between the projective Orlik-Solomon algebra of a matroid and the tropical cohomology of its Bergman fan, linking algebraic and geometric structures.

Contribution

It demonstrates a fundamental connection between the Orlik-Solomon algebra and tropical cohomology of the Bergman fan of a matroid, revealing a new algebraic-geometric correspondence.

Findings

The projective Orlik-Solomon algebra is isomorphic to the tropical cohomology of the Bergman fan.

Establishes a canonical isomorphism linking algebraic and geometric structures in matroid theory.

Bridges combinatorial matroid invariants with tropical geometric concepts.

Abstract

Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $\Sigma_0(M)$. On the other hand to any rational polyhedral fan $\Sigma$ one can associate its tropical homology and cohomology groups $\F_\bullet(\Sigma)$, $\F^\bullet (\Sigma)$. We will show that the projective Orlik-Solomon algebra $OS_0(M)$ is canonically isomorphic to $\F^\bullet (\Sigma_0(M))$.