The Singular Cohomology Ring of a Matroid

TL;DR

This paper introduces a new singular cohomology ring for matroids, extending the Chow ring, and explores its properties using toric varieties, matroidal flips, and building sets, revealing new algebraic and combinatorial insights.

Contribution

It defines the singular cohomology ring of a matroid via associated toric varieties and generalizes it to arbitrary building sets, providing new tools for matroid theory.

Findings

Proves vanishing results for the cohomology ring.

Computes the dimension of top-weight cohomology using the Möbius invariant.

Provides a recursive formula for Hodge numbers of uniform matroids.

Abstract

We introduce the singular cohomology ring of a matroid which extends the Chow ring of a matroid. This is defined as the singular cohomology ring of a certain quasi-projective toric variety associated to the matroid. Using the matroidal flips of Adiprasito, Huh, and Katz, we prove sharp vanishing results for the cohomology ring and compute the dimension of the top-weight cohomology in terms of the M\"{o}bius invariant of the matroid. In the case of uniform matroids, these techniques give a recursive formula for the Hodge numbers. Finally, we generalize the singular cohomology ring to arbitrary building sets on the lattice of flats, and we show how the cohomology depends on the building set.