Focal matroids of covers and homological properties of matroids

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.

💡 Research Summary

This paper establishes that the Stanley–Reisner ideal (or equivalently the cover ideal) of any matroid admits a minimal free resolution obtained by iterated mapping cones, a result that had previously been known only for monomial ideals with linear quotients. The authors introduce a novel combinatorial construction called “focal matroids.” For a simplicial complex Δ and an integer ℓ, an ℓ‑cover is a function γ: