Polarizations of Artin monomial ideals

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.