Complete modules and torsion modules

Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded R-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and A = [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /].