Completion and torsion over commutative DG rings

Let CDG_cont be the category whose objects are pairs ((Aa¯)), where A is a commutative DG-algebra and a¯ ⊆ H (A) is a finitely generated ideal, and whose morphisms f(Aa¯) → (Bb¯) are morphisms of DG-algebras A → B, such that (H (f) (a¯)) ⊆ b¯. Letting Ho(CDG_cont) be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor LΛ: Ho(CDG_cont) → Ho(CDG_cont) which takes a pair ((Aa¯)) into its non-abelian derived a¯ -adic completion. We show that this operation has, in a derived sense, the usual properties of adic completion of commutative rings, and that if A = H⁰(A) is an ordinary noetherian ring, this operation coincides with ordinary adic completion. As an application, following a question of Buchweitz and Flenner, we show that if k is a commutative ring, and A is a commutative k-algebra which is a-adically complete with respect to a finitely generated ideal a⊆ A, then the derived Hochschild cohomology modules ExtA⊗kLAn(AA) and the derived complete Hochschild cohomology modules ExtA⊗^kLAn(AA) coincide, without assuming any finiteness or noetherian conditions on k, A or on the map k→ A.