Periodic cyclic homology

General Introduction
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Online References
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Paper References
Book by Meyer:
Periodic cyclic homology is a homology theory for noncommutative algebras that plays a similar role in noncommutative 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 Ktheory, and the Chernâ€“Connes character for Ktheory and Khomology.
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 noncommutative bornological algebras.
The book is mainly intended for researchers and advanced graduate students interested in noncommutative 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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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Cortinas: Periodic cyclic homology as sheaf cohomology: K0307
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Other Information
arXiv:1003.3210 Motivic structures in noncommutative 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 noncommutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an RHodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.
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