The Singular Cohomology Ring of a Matroid

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 Möbius 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.