Let A be a ring and let I ā A be a finitely generated ideal. The local cohomology functor and derived completion functor define quasi-inverse equivalences of categories D_{Iā-torsion}(A) <---> D_{comp}(A,I). See Proposition Tag 0A6X.

This and more general statements can be found in Dwyer-Greenlees. As usual more references are welcome. Thanks!