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(greater than or equal to0) --> ab(Delta) 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(Delta), although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and Rings(Delta) are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* (similar to)--> Ho Rings(Delta). 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* --> Rings(Delta), 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] --> Pi(R)(n) S that arises R from a homomorphism R --> S of not necessarily commutative rings, using the coproduct PiA(R) 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 S-R(circle timesn) originally defined by Nuss K-theory 12 (1997) 23, using braids. (C) 2003 Elsevier B.V. All rights reserved.