On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology

Abstract

In section 1. the notion of (strong) nil- homotopy is introduced, and its first properties are proved. Section 2 is devoted to the interpretation of quasi-free pro-algebras as cofibrant objects with respect to the setting of the previous section (Theorem 2.1). The notion of weak nil-homotopy is introduced in section 3, where the existence of the localized category Def lPA is proved (Theorem 1. A Closed Model Category Analogy. 1.0 We consider associative, non-necessarily unital algebras over a fixed ground field k. We write A and V for the categories of algebras and vector spaces and PA and VV for the corresponding pro-categories. As in [CQ3] we consider only countably indexed pro-objects. A map f E PA(A.B) is called a fibration if it admits a right inverse as a map of pro-vector spaces, i.e. there exists s E PV(B.A) such that fs = 1. Fibrations are denoted by a double headed arrow By a (nil-) deformation (-») of a pro-algebra 3.2) . Section 2 is devoted to the comparison between our notion of nil-homotopv and the usual, polynomial homotopy. We prove that the localization at the union of the classes of nil-deformations and graded deformations exists and can be calculated as a homotopy category (Theorem 4.1). Section 5 deals with the formalization of the derived functor analogy of [CQ2]. We establish sufficient conditions for the existence of left derived functors (Theorem 5.2) and prove that, in characteristic zero, these conditions are met by the de Rham supercoinplex functor A θ- XA of Cuntz-Quillen (Corollary 5.4). In section 6 we compute the derived functor of the rational A’-theorv of rational pro-algebras. (Theorem 6.2) and of the negative cyclic homology of pro-algebras over any field (Corollary 6.9).