Abstrakt
Let CDG(cont) be the category whose objects are pairs (A, (a) over bar), where A is a commutative DG-algebra and (a) over bar subset of H-0(A) is a finitely generated ideal, and whose morphisms f : (A, (a) over bar) -> (B, (b) over bar) are morphisms of DG-algebras A -> B, such that (H0(f)((a) over bar)) subset of (b) over bar. Letting Ho(CDG(cont)) be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor L. : Ho(CDG(cont)) -> Ho(CDG(cont)) which takes a pair (A, (a) over bar) into its non-abelian derived (a) over bar -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 = H-0(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 subset of A, then the derived Hochschild cohomology modules Ext(A circle times LkA)(n) (A, A) and the derived complete Hochschild cohomology modules Ext(A (circle times) over cap LkA)(n) (A, A) coincide, without assuming any finiteness or noetherian conditions on k, A or on the map k -> A.