  Let $U$ be a smooth connected complex algebraic variety, and let $f\colon
U\to \mathbb C^*$ be an algebraic map. To the pair $(U,f)$ one can associate an
infinite cyclic cover $U^f$, and (homology) Alexander modules are defined as
the homology groups of this cover. In two recent works, the first of which is
joint with Geske, Maxim and Wang, we developed two different ways to put a
mixed Hodge structure on Alexander modules. Since they are not finite
dimensional in general, each approach replaces the Alexander module by a
different finite dimensional module: one of them takes the torsion submodule,
the other takes finite dimensional quotients, and the constructions are not
directly comparable. In this note, we show that both constructions are
compatible, in the sense that the map from the torsion to the quotients is a
mixed Hodge structure morphism.
