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The Singular Cohomology Ring of a Matroid

Kyle Binder

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 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.

The Singular Cohomology Ring of a Matroid

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 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.

Paper Structure

This paper contains 21 sections, 37 theorems, 118 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Background and notation
  5. Matroids
  6. Basis activity
  7. Bergman fans of order filters
  8. Bergman fans of building sets
  9. Toric varieties
  10. Singular cohomology of smooth toric varieties
  11. Koszul homology
  12. The singular cohomology ring of a matroid
  13. The Hodge--Poincaré polynomial of the lifted Bergman fan of the empty order filter via Hochster's formula
  14. The Hodge--Poincaré polynomial of the Bergman fan of the empty order filter. via flat base change
  15. Vanishing results for the singular cohomology ring of a matroid
  16. ...and 6 more sections

Key Result

Theorem 2.18

For a smooth toric variety $X_{\Sigma}$, there is an isomorphism of rings where $\mathbb{Q}$ is the residue field of the maximal homogeneous ideal.

Figures (4)

  • Figure 1: The lattice of flats of the broom matroid$M^{\mathrm{br}}$, and various Bergman fans. The bottom-most simplex of $\Sigma_{M^{\mathrm{br}}, \emptyset}$ is filled in, so the fan is not pure.
  • Figure 2: The Bergman fan of the empty order filter and the Bergman fan for $U_{3,4}$.
  • Figure 3: The Bergman fan $\Sigma_{U_{2,3}}$.
  • Figure 4: $\Sigma_{U_{3,3}, \emptyset}$ and the stellar subdivision along $\tau = \mathop{\mathrm{\mathop{pos}}}\nolimits(\rho_{1}, \rho_{2})$.

Theorems & Definitions (106)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 96 more