  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.
