Linear Strands Supported on Regular CW Complexes

TL;DR

This paper investigates the conditions under which the linear strands of certain monomial ideals can be supported on regular CW complexes, providing new criteria and applying them to classes like rainbow monomial ideals and initial ideals of minors.

Contribution

It introduces new sufficient conditions for linear strands to be supported on CW complexes and characterizes when these ideals have linear resolutions, including applications to polarizations.

Findings

Rainbow monomial ideals have linear strands supported on regular CW complexes.

Certain initial ideals of maximal minors are supported on CW complexes.

Conditions for linear resolutions of these ideals are established.

Abstract

In this paper, we study ideals $I$ whose linear strand can be supported on a regular CW complex. We provide a sufficient condition for the linear strand of an arbitrary subideal of $I$ to remain supported on an easily described subcomplex. In particular, we prove that a certain class of rainbow monomial ideals always have linear strand supported on a regular CW complex, including any initial ideal of the ideal of maximal minors of a generic matrix. We also provide a sufficient condition for these ideals to have linear resolution, which is also an equivalence under mild assumptions. We then employ a result of Almousa, Fl\o ystad, and Lohne to apply these results to polarizations of Artinian monomial ideals. We conclude with further questions relating to cellularity of certain classes of squarefree monomial ideals and the relationship between initial ideals of maximal minors and algebra structures on certain resolutions.