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

Abstract

The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ 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: LK: Ho DGR* Ho RingsΔ. 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] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐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⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids.