Cohochschild and Cocyclic Homology of Chain Coalgebras

Generalizing work of Doi and of Farinati and Solotar, we define coHochschild and cocyclic homology theories for chain coalgebras over any commutative ring and prove their naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex admits a natural comultiplicative structure. We show that a twisting cochain from a chain coalgebra to a chain algebra naturally induces a chain map from the coHochschild and cocyclic complexes of the coalgebra to the Hochschild and cyclic complexes of the algebra and determine conditions under which the induced maps in homology are isomorphisms. The coHochschild complex is topologically relevant as well. Given two simplicial maps g, h : K → L, the homology of the coHochschild complex of C∗L with coefficients in C∗K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. Introduction Hochschild and cyclic homology are well-known and very useful homology theories for algebras, which have considerable relevance in topology as well. In particular, for any based topological space X , the Hochschild homology of S∗(ΩX), the singular chains on the space of based loops on X , is isomorphic to the singular homology of the space LX of free loops on X , while the cyclic homology of S∗(ΩX) is isomorphic to the S-equivariant homology of LX (cf, e.g., [18, 20]). In [4] Doi developed a homology theory for coalgebras over a field that is analogous to the Hochschild homology of algebras. Farinati and Solotar later extended Doi’s theory, defining a homology theory analogous to cyclic homology of algebras, first for ungraded coalgebras over a field [7], then for chain coalgebras over a field [6]. In this article, we offer an alternate approach to the homology theories of Doi and of Farinati and Solotar, which allows us to extend their theories, which we call coHochschild and cocyclic homology, easily to chain coalgebras over any commutative ring. We provide descriptions Date: November 20, 2008. 2000 Mathematics Subject Classification. Primary: 16E40, 19D55 Secondary: 18G60, 55M20, 55U10, 81T30.