Triplets of pure free squarefree complexes

TL;DR

This paper explores special triplets of pure free squarefree complexes over polynomial rings, conjecturing their existence and uniqueness, and provides constructions under certain linearity conditions.

Contribution

It introduces the concept of triplets of pure free squarefree complexes, conjectures their existence and uniqueness, and constructs examples when two complexes are linear.

Findings

Uniqueness of Betti numbers follows from existence.

Constructs triplets when two complexes are linear.

Proposes conjectures on existence and uniqueness of such triplets.

Abstract

On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an endofunctor on this category, of order three up to translation. We consider complexes F of free squarefree modules such that both F, AD(F) and (AD)^2(F) are pure, when considered as singly graded complexes. We conjecture i) the existence of such triplets of complexes for given triplets of degree sequences, and ii) the uniqueness of their Betti numbers, up to scalar multiple. We show that this uniqueness follows from the existence, and we construct such triplets if two of them are linear.