Cosimplicial versus Dg-rings: a Version of the Dold-kan Correspondence

The (dual) Dold-Kan correspondence says that there is an equivalence of categories K : Ch → Ab between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR → Rings, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR and Rings are Quillen closed model categories and the left derived functor of K is an equivalence: LK : HoDGR ∼ −→ HoRings The dual of this result for chain DG and simplicial rings was obtained independently by S. Schwede and B. Shipley through different methods (Equivalences of monoidal model categories. Algebraic and Geometric Topology 3 (2003), 287-334). Our proof is based on a functor Q : DGR → Rings, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-KonstantRosenberg and Loday-Quillen theorems. Our version applies to the cyclic module that arises from a homomorphism R → S of not necessarily commutative rings when the coproduct ∐ R of associative R-algebras is substituted for ⊗R. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S n R originally defined by P. Nuss using braids (Noncommutative descent and nonabelian cohomology, K-theory 12 (1997) 23-74.).