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 $K:\cha\to \Ab^Δ$ 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 $DGR^*\to\ass^Δ$, although not itself an equivalence, does induce one at the level of homotopy categories. The dual of this result for chain $DG$ and simplicial rings was obtained independently by S. Schwede and B. Shipley through different methods ({\it Equivalences of monoidal model categories}. Algebraic and Geometric Topology 3 (2003), 287-334). Our proof is based on a functor $Q:DGR^*\to \ass^Δ$, naturally homotopy equivalent to $K$, 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-Konstant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module that arises from a homomorphism $R\to S$ of not necessarily commutative rings when the coproduct $\coprod_R$ of associative $R$-algebras is substituted for $\otimes_R$. As another application of the properties of $Q$, we obtain a simple, braid-free description of a product on the tensor power $S^{\otimes_R^n}$ originally defined by P. Nuss using braids ({\it Noncommutative descent and nonabelian cohomology,} K-theory {\bf 12} (1997) 23-74.).