Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

TL;DR

This paper generalizes the concept of chordal graphs to higher-dimensional simplicial complexes, providing new conditions for when monomial ideals have linear resolutions, especially over fields of characteristic 2.

Contribution

It introduces d-chorded and orientably-d-cycle-complete complexes, extending Fr"oberg's theorem to higher dimensions and establishing a necessary combinatorial condition for componentwise linearity.

Findings

d-dimensional trees correspond to ideals with linear resolutions over characteristic 2

Introduces higher-dimensional chordal complexes

Provides a necessary condition for componentwise linearity

Abstract

In this paper we extend one direction of Fr\"oberg's theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.