On basic double G-links of squarefree monomial ideals

TL;DR

This paper investigates basic double G-links of squarefree monomial ideals, extending beyond weakly vertex decomposable complexes, and provides structural insights and examples related to Cohen-Macaulay complexes not arising from such links.

Contribution

It offers a structural characterization of basic double G-links involving edge ideals and demonstrates that certain Cohen-Macaulay complexes are not basic double links.

Findings

Basic double G-link involving edge ideals must be of degree 1.

Provides a generating set for the ideal involved in the G-link.

Examples of Cohen-Macaulay complexes are not basic double links.

Abstract

Nagel and R\"omer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley-Reisner ideal of every weakly vertex decomposable simplicial complex is Gorenstein linked to an ideal of indeterminates via a sequence of basic double G-links. In this paper, we explore basic double G-links between squarefree monomial ideals beyond the weakly vertex decomposable setting. Our first contribution is a structural result about certain basic double G-links which involve an edge ideal. Specifically, suppose $I(G)$ is the edge ideal of a graph $G$. When $I(G)$ is a basic double G-link of a monomial ideal $B$ on an arbitrary homogeneous ideal $A$, we give a generating set for $B$ in terms of $G$ and show that this basic double G-link must be of degree $1$. Our second focus is on examples from the literature of simplicial complexes known to be Cohen-Macaulay but not weakly vertex decomposable. We show that these examples are not basic double links of any other squarefree monomial ideals.