On The Image Of The Totalling Functor

Let A denote a DG algebra and k a field. The totalling functor, from the category of chain complexes over the graded A-modules to the catagory of DG modules over A, can be extended to one between their derived categories. If this extension were onto, the derived category of the category of DG modules would be superfluous. This paper investigates the image of the extension of Tot in the fundamental case when A is the polynomial ring in d variables over k. When d is at least 2, there are semifree DG modules of rank n, where n is at least 4, that are not obtained from the totalling of any complex of graded A-modules. However, when A=k[x], every rank n semifree DG module over A is in the image of Tot. Moreover, for a polynomial ring of arbitrary size, we will define a special class of rank n semifree DG modules over A which are always in the image of Tot.