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

TL;DR

This paper formalizes the analogy between periodic cyclic homology and derived functors of de Rham cohomology in the Cuntz-Quillen framework, extending it to all characteristics and computing related derived functors.

Contribution

It establishes the existence of a localization of pro-algebras and shows periodic cyclic homology as a derived functor of de Rham cohomology in characteristic zero.

Findings

Localization of pro-algebras at infinitesimal deformations exists in any characteristic.

Periodic cyclic homology is the derived functor of de Rham cohomology in characteristic zero.

Derived functor of rational K-theory is the fiber of the Chern character to negative cyclic homology.

Abstract

Cuntz and Quillen have shown that for algebras over a field $k$ with $char(k)=0$, periodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to formalize this derived functor analogy. We show that the localization ${Def}^{-1}\Cal{PA}$ of the category $\Cal{PA}$ of countable pro-algebras at the class of (infinitesimal) deformations exists (in any characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic homology is the derived functor of de Rham cohomology with respect to this localization (Corollary 5.4). We also compute the derived functor of rational $K$-theory for algebras over $\Bbb Q$, which we show is essentially the fiber of the Chern character to negative cyclic homology (Theorem 6.2).