Chow rings of matroids are Koszul

TL;DR

This paper proves Dotsenko's conjecture that the Chow ring of any matroid is Koszul, establishing important algebraic properties that have implications for combinatorics and algebraic geometry.

Contribution

The paper proves that the Chow ring and augmented Chow ring of any matroid are Koszul, confirming a longstanding conjecture and advancing understanding of their algebraic structure.

Findings

Chow rings of matroids are Koszul.

Augmented Chow rings of matroids are Koszul.

Chow rings have rational Poincaré series.

Abstract

Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. Dotsenko conjectured that the Chow ring of any matroid is Koszul. The purpose of this paper is to prove Dotsenko's conjecture. We also show that the augmented Chow ring of a matroid is Koszul. As a corollary, we show that the Chow rings and augmented Chow rings of matroids have rational Poincar\'{e} series.