Graded Linearity of Stanley-Reisner Ring of Broken Circuit Complexes

TL;DR

This paper develops new notions of graded linearity for modules over polynomial rings, compares them with existing concepts, and applies these ideas to Stanley-Reisner rings of broken circuit complexes and hyperplane arrangements, revealing structural decompositions.

Contribution

It introduces graded linear resolution and quotients, compares them with componentwise linearity, and characterizes these properties for Stanley-Reisner rings of broken circuit complexes, extending matroid decomposition results.

Findings

Graded linear quotients imply graded linear resolution for certain modules.

Matroids can be stratified into parts decomposable into uniform matroids.

Results extend to Orlik-Terao ideals of hyperplane arrangements.

Abstract

This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of $A$, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley-Reisner ring of broken circuit complexes and generalize several results of \cite{RV} on the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each strata has a decomposition into uniform matroids. We also present analogs of our results for the Orlik-Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.