On the Image of the Totaling Functor

TL;DR

This paper studies the totaling functor from complexes of graded modules to DG modules over a DG algebra, identifying classes of modules that can be expressed as totalings, especially over polynomial rings.

Contribution

It characterizes the image of the totaling functor for DG algebras with trivial differential, including polynomial rings, and identifies semifree DG modules that are totalings.

Findings

Certain semifree DG modules are always totalings of complexes of graded free modules.

The image of the totaling functor is characterized for polynomial rings over a field.

Results provide a deeper understanding of the structure of DG modules over specific DG algebras.

Abstract

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.