Linear resolutions and quasi-linearity of monomial ideals

TL;DR

This paper introduces the concept of quasi-linearity in monomial ideals, establishing its necessity for linear resolutions and characterizing quadratic monomial ideals with this property.

Contribution

It defines quasi-linearity and strongly linear monomials, providing criteria for when monomial ideals have linear resolutions based on these concepts.

Findings

Quasi-linearity is necessary for a monomial ideal to have a linear resolution.

Characterization of all quasi-linear quadratic monomial ideals.

A criterion involving strongly linear monomials for linear resolutions.

Abstract

We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal $I$ and prove that if $\mathbf{u}$ is a strongly linear monomial over $I$ then $I$ has a linear resolution (resp: is quasi-linear) if and only if $I+\mathbf{u}\mathfrak{p}$ has a linear resolution (resp: is quasi-linear). Here $\mathfrak{p}$ is any monomial prime ideal.