Polarizations of Artin monomial ideals

TL;DR

This paper proves that polarizations of Artin monomial ideals produce triangulated balls, confirming a conjecture and linking geometric and algebraic properties of these ideals.

Contribution

It establishes that all polarizations of Artin monomial ideals form triangulated balls and classifies their structures in certain cases.

Findings

Polarizations of Artin monomial ideals define triangulated balls.

Full-dimensional Cohen-Macaulay sub-complexes are of this kind.

Squeezed balls derive from these polarizations.

Abstract

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals.