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

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

2000-01-31

arxiv.org/abs/math/0001179v1

Algebra Colloquium 5(3) (1998), 305-328

Mathematics

K-Theory and Homology

22 pages, AmsTex, uses xy.tex for commutative diagrams

Scientific paper

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).

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