On the Image of the Totaling Functor

Let $A$ be a DG algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of graded $A$-modules to the category of DG $A$-modules. Specifically, we exhibit a special class of semifree DG $A$-modules which can always be expressed as the totaling of some complex of graded free $A$-modules. As a corollary, we also provide results concerning the image of the totaling functor when $A$ is a polynomial ring over a field.