Ordinary and symbolic powers of matroids via polarization

TL;DR

This paper introduces a unified approach using polarization to analyze the algebraic properties of powers of matroid-related squarefree monomial ideals, simplifying proofs of key results in combinatorial commutative algebra.

Contribution

It provides a new, elementary method to recover and extend known results on Cohen-Macaulayness and Serre's conditions for powers of matroid ideals, with simplified proofs.

Findings

Unified polarization approach for squarefree monomial ideals

Elementary proofs of Cohen-Macaulay and Serre's conditions

Characterization of algebraic properties via matroid structure

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.