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Ordinary and symbolic powers of matroids via polarization

Justin Lyle, Paolo Mantero

TL;DR

This work develops a uniform polarization-based framework to study squarefree monomial ideals and their powers, connecting combinatorial matroid structure to algebraic properties via the naive dual $ riangle^{(m)}$ and its polarization. By coupling this approach with Hochster–Huneke graphs, the authors give short proofs of key results on when symbolic and ordinary powers are Cohen–Macaulay or satisfy Serre's $(S_2)$, and they extend these characterizations to new equivalent conditions. Notable contributions include a concise proof of the Minh–Trung/Varbaro–Terai–Trung characterizations, a new set of equivalences for symbolic and ordinary powers of Stanley–Reisner ideals of matroids, a closed formula for the regularity of symbolic powers, and a level-property dichotomy tied to circuit sizes. The results illuminate how matroidal combinatorics tightly constrain algebraic properties across all powers, with additional insights into glicci status of polarizations and mixed symbolic powers, offering streamlined proofs and avenues for further exploration in combinatorial commutative algebra.

Abstract

In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goal: (i) recover and extend several well--known results in the literature, especially regarding Stanley--Reisner ideals of matroids, and (ii) provide short, elementary proofs for these results. Among them, we provide simple proofs of two celebrated results of Minh and Trung, Varbaro, and Terai and Trung elegantly characterizing the Cohen-Macaulay property, or even Serre's condition $(S_2)$, of symbolic and ordinary powers of squarefree monomial ideals in terms of their combinatorial (matroidal) structure. Our work relies on the interplay of several combinatorial and algebraic concepts, including dualities, polarizations, Serre's conditions, matroids, Hochster-Huneke graphs, vertex decomposability, and careful choices of monomial orders.

Ordinary and symbolic powers of matroids via polarization

TL;DR

This work develops a uniform polarization-based framework to study squarefree monomial ideals and their powers, connecting combinatorial matroid structure to algebraic properties via the naive dual and its polarization. By coupling this approach with Hochster–Huneke graphs, the authors give short proofs of key results on when symbolic and ordinary powers are Cohen–Macaulay or satisfy Serre's , and they extend these characterizations to new equivalent conditions. Notable contributions include a concise proof of the Minh–Trung/Varbaro–Terai–Trung characterizations, a new set of equivalences for symbolic and ordinary powers of Stanley–Reisner ideals of matroids, a closed formula for the regularity of symbolic powers, and a level-property dichotomy tied to circuit sizes. The results illuminate how matroidal combinatorics tightly constrain algebraic properties across all powers, with additional insights into glicci status of polarizations and mixed symbolic powers, offering streamlined proofs and avenues for further exploration in combinatorial commutative algebra.

Abstract

In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goal: (i) recover and extend several well--known results in the literature, especially regarding Stanley--Reisner ideals of matroids, and (ii) provide short, elementary proofs for these results. Among them, we provide simple proofs of two celebrated results of Minh and Trung, Varbaro, and Terai and Trung elegantly characterizing the Cohen-Macaulay property, or even Serre's condition , of symbolic and ordinary powers of squarefree monomial ideals in terms of their combinatorial (matroidal) structure. Our work relies on the interplay of several combinatorial and algebraic concepts, including dualities, polarizations, Serre's conditions, matroids, Hochster-Huneke graphs, vertex decomposability, and careful choices of monomial orders.

Paper Structure

This paper contains 15 sections, 31 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Complete intersections
  4. Cohen--Macaulayness and Serre's conditions
  5. Matroids
  6. Vertex decomposable simplicial complexes
  7. Polarizations
  8. Hochster--Huneke graphs
  9. The Cohen-Macaulay Property and Serre's Condition for Symbolic Powers
  10. The proof of Theorem \ref{['cmsymbintro']}
  11. Application 1. The simplicial complexes $\Delta^{(m)}$ are glicci
  12. Application 2. Mixed symbolic powers of $I_\Delta$
  13. Ordinary powers
  14. Regularity of symbolic powers
  15. The Level Property

Key Result

Theorem 1.1

This is how the theorem is stated in CN76. However, it is not hard to see that their proof essentially proves the following more general statement: Let $R$ be a Cohen--Macaulay positively graded $k$-algebra, and let $I$ be a graded ideal that is generically a complete intersection. Then $I^m$ is Coh

Theorems & Definitions (71)

  • Theorem 1.1: Cowsik--Nori 1976, Dade 1960
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 61 more