Polarizations of Artin monomial ideals

Abstract

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This settles a conjecture of A. Almousa, H. Lohne and the first author [3]. Geometrically, polarizations of ideals strictly containing ( x 1 a 1 , ... , 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 , ⋯ , 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 , ⋯ , a n - 1 . We prove that this cell complex gives a cellular minimal free resolution 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 of examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G. Kalai [32] derive from polarizations of Artin monomial ideals.