Cosimplicial versus DGrings: a version of the DoldKan correspondence
Abstract
The (dual) DoldKan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^\Delta$ 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 $DGR^*\to\ass^\Delta$, 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), 287334). Our proof is based on a functor $Q:DGR^*\to \ass^\Delta$, 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 HochschildKonstantRosenberg and LodayQuillen 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, braidfree 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,} Ktheory {\bf 12} (1997) 2374.).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2003
 arXiv:
 arXiv:math/0306289
 Bibcode:
 2003math......6289C
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Topology;
 18G55
 EPrint:
 Final version to appear in JPAA. Large parts rewritten, especially in the last section.Proof of main theorem simplified