On the homology of completion and torsion

Let A be a commutative ring, and a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically a -adically complete complexes and the category of cohomologically a -torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso-Jeremias-Lipman, Schenzel and Dwyer-Greenlees.