On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology, by Guillermo Cortiñas, http://www.math.uiuc.edu/K-theory/0223
Cuntz and Quillen have shown that, if char(k)=0, then 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 of the category of countably indexed pro-algebras at the class of deformations exists, and that periodic cyclic homology is the derived functor of de Rham (co)homology with respect to this localization. We also compute the derived functor of rational K-theory, and show it is essentially the fiber of the rational Jones-Goodwillie map to negative cyclic homology.
This paper has appeared in Alg. Colloquium vol. 5, no. 3 (1998) 305-328, so has been removed from this server at the request of the author.
Cortinas: Periodic cyclic homology as sheaf cohomology: http://www.math.uiuc.edu/K-theory/0307
http://www.math.uiuc.edu/K-theory/0349
Book by Meyer:
Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book the author develops and compares these theories, emphasising their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to K-theory, and the Chern–Connes character for K-theory and K-homology.
The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras.
The book is mainly intended for researchers and advanced graduate students interested in non-commutative geometry. Some chapters are more elementary and independent of the rest of the book, and will be of interest to researchers and students working in functional analysis and its applications.
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NCG (Algebra and noncommutative geometry)
arXiv:1003.3210 Motivic structures in non-commutative geometry from arXiv Front: math.AG by D. Kaledin We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional “motivic” structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.
nLab page on Periodic cyclic homology