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Focal matroids of covers and homological properties of matroids

Paolo Mantero, Vinh Nguyen

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.

Focal matroids of covers and homological properties of matroids

Abstract

In this paper we prove that the Stanley--Reisner ideal or cover ideal 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 that are constructed relative to minimal -covers of . Our second main result is that the monomial support of the multigraded Betti numbers of corresponds precisely to the squarefree minimal generators of the symbolic powers of . 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 are minimally resolvable by iterated mapping cones.
Paper Structure (9 sections, 28 theorems, 34 equations)

This paper contains 9 sections, 28 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and basic facts
  4. Symbolic powers of $C$-matroidal ideals
  5. Standard Monomial Decomposition
  6. Restriction, Contraction, and Colon Ideals
  7. focal matroids and their structure
  8. Iterated Contractions and Iterated Mapping Cones
  9. Resolution of $C$-matroidal Ideals and Formulas for Homological Invariants

Key Result

Proposition 2.2

Let $\mathcal{M}$ be a matroid on $[n]$. Then

Theorems & Definitions (69)

  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Remark 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • Lemma 2.11
  • ...and 59 more