Focal matroids of covers and homological properties of matroids

TL;DR

This paper explores the homological properties of matroids through the lens of their Stanley--Reisner ideals, introducing focal matroids and establishing a new characterization of matroidal ideals via Betti numbers.

Contribution

It introduces focal matroids and proves that the Stanley--Reisner ideal of a matroid is minimally resolvable by iterated mapping cones, providing new homological insights.

Findings

Stanley--Reisner ideal of a matroid is minimally resolvable by iterated mapping cones

The multigraded Betti numbers' support matches minimal generators of symbolic powers

Matroidal ideals are uniquely characterized by their Betti number support

Abstract

In this paper we prove that the Stanley--Reisner ideal or cover ideal $I$ of a matroid is minimally resolvable by iterated mapping cones. As a technical tool for this purpose, we introduce and study focal matroids, which are submatroids of a matroid $\mathcal{M}$ that are constructed relative to minimal $\ell$-covers of $\mathcal{M}$. Our second main result is that the monomial support of the multigraded Betti numbers of $I$ corresponds precisely to the squarefree minimal generators of the symbolic powers of $I$. In fact, we prove that matroidal ideals are the only squarefree ideals with this property, thus obtaining a new homological characterization of matroidal ideals. These techniques are foundational for a follow-up paper, where we will show that all symbolic power of $I$ are minimally resolvable by iterated mapping cones.