Monomial ideals with 3-linear resolutions

TL;DR

This paper investigates the regularity of square-free monomial ideals generated in degree 3, introducing operations on associated clutters that preserve regularity and identifying classes of ideals with linear resolutions.

Contribution

It defines operations on clutters that preserve regularity and characterizes when certain clutters have linear resolutions, including triangulations of spheres.

Findings

Regularity is conserved under specific clutter operations.

Certain classes of ideals with linear resolutions are introduced.

Triangulations of spheres do not have linear resolutions, but their sub-clutters do.

Abstract

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these operations. We apply the operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper sub-clutter of it has a linear resolution.