Cosimplicial versus DG-rings: a version of the Dold–Kan correspondence

The (dual) Dold–Kan correspondence says that there is an equivalence of categories image 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*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence:image The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, 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–Kostant–Rosenberg and Loday–Quillen theorems. Our version applies to the cyclic module [n]maps tocoproduct operatornRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct coproduct operatorR of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power Scircle times operatorRn originally defined by Nuss K-theory 12 (1997) 23, using braids.