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

JL. CASTIGLIONI; G. CORTIÑAS

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Año: 2004 vol. 191 p. 119 - 142

0022-4049

The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:cha o Ab^Delta$ 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 $DGR^* oass^Delta$, 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 $Q:DGR^* o ass^Delta$, 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 $R o S$ 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 {f 12} (1997).