Three homotopy theories for cyclic modules

Abstract

1.1. The main result. Let R be a ring with 1 +O and let C denote the category of the cyclic R-modules of Connes, i.e., [3, 61 simplicial R-modules X with, in each dimension n LO, an extra degeneracy map s,+, : X,+X,,+, satisfying the usual identities, except that the identities des,+, = ,, ,, s d : X,-X,, (n LO) are replaced by (dos,+,)“+‘= id :X,-X, (n10). Such a cyclic R-module has not only its underlying simplicial R-module, which we will denote by j *X, but also an underlying cosimplicial R-module k*X (with di=s,_i: X,,_i+Xn and s’=d,_i: Xn+XnPi). As a result, there are three obvious homotopy theories which one can associate with C; these correspond to three possible criteria for calling a map f: X-+X’ in C a ‘weak equivalence’ [6, Q 71. First there is the one-sided homotopy theory in which the weak equivalences are the maps f which induce isomorphisms xi j *X= n, j *X (ilO) on the homotopy groups of the underlying simplicial modules. Next, there is a dual one-sided theory in which the weak equivalences are the maps f which induce isomorphisms ?k*Xn’k*X’ (ir0) on the cohomotopy groups of the underlying cosimplicial modules. Finally, there is a strong or two-sided theory in which the weak equivalences are the maps f which induce isomorphisms on both the homotopy groups of the underlying simplicial modules and the cohomotopy groups of the underlying cosimplicial modules. The main aim of this paper is to show that each of these three homotopy theories is equivalent to a corresponding homotopy theory of differential graded modules over a graded exterior R-algebra. This implies that from any of the above three points of view the study of cyclic R-modules is equivalent to the study of the classical homological algebra of certain chain complexes. In more detail: