Chordality, $d$-collapsibility, and componentwise linear ideals

TL;DR

This paper introduces a new class of chordal simplicial complexes based on $d$-collapsibility, demonstrating their associated Stanley-Reisner ideals are componentwise linear, thus linking combinatorial topology with algebraic properties of ideals.

Contribution

It extends the concept of chordal clutters to $d$-collapsible complexes, establishing their connection to componentwise linear ideals and Betti table characterization.

Findings

$d$-collapsible complexes produce componentwise linear ideals.

Chordal complexes include Götzmann and square-free stable ideals.

Betti tables of all componentwise linear ideals match those of chordal complexes.

Abstract

Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal clutters'' which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with linear resolution in a polynomial ring. We show $d$-collapsible and $d$-representable complexes produce componentwise linear ideals for appropriate $d$. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees. We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.