Homologie cyclique et K-théorie / Max Karoubi

Auteur: Karoubi, Max (1938-) - AuteurType de document: MonographieCollection: Astérisque ; 149Langue: françaisPays: FranceÉditeur: Paris : Société Mathématique de France, 1987Description: 1 vol. (147 p.) : ill. ; 24 cmNote: This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independently from the work of Alain Connes and the author. They were motivated by considerations of operator algebras and algebraic K-theory. The good framework to develop these considerations is cyclic homology (or non-commutative De Rham homology). This enables us to extend this classical Chern-Weil theory far beyond its original scope (at least for the general linear group). Cyclic homology is the natural target for characteristic classes and its computation is a matter of homological algebra as was shown by A. Connes. On the other hand, the objects we are taking the characteristic classes of are elements of the K-theory of a ring. This K-theory (algebraic or topological) is difficult to compute in general. Cyclic homology (also called “additive K-theory” by Feigin and Tsygan) appears therefore as a first step to compute the K- groups for general algebras. We have tried to make this book as self- contained as possible. Together with the motivations provided by A. Connes, the reader should not find any special difficulty to read it. In particular, the first chapters can be easily integrated in a graduate course on the subject. (Zentralblatt)Bibliographie: Bibliogr. p. 140-142. Index. Sujets MSC: 18F25 Category theory; homological algebra -- Categories and geometry -- Algebraic K-theory and L-theory
55R40 Algebraic topology -- Fiber spaces and bundles -- Homology of classifying spaces, characteristic classes
55R50 Algebraic topology -- Fiber spaces and bundles -- Stable classes of vector space bundles, K-theory
53C05 Differential geometry -- Global differential geometry -- Connections, general theory
En-ligne: Numdam
Location Call Number Status Date Due
Couloir 09662-01 / Séries SMF 149 (Browse Shelf) Available

This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independently from the work of Alain Connes and the author. They were motivated by considerations of operator algebras and algebraic K-theory. The good framework to develop these considerations is cyclic homology (or non-commutative De Rham homology). This enables us to extend this classical Chern-Weil theory far beyond its original scope (at least for the general linear group). Cyclic homology is the natural target for characteristic classes and its computation is a matter of homological algebra as was shown by A. Connes. On the other hand, the objects we are taking the characteristic classes of are elements of the K-theory of a ring. This K-theory (algebraic or topological) is difficult to compute in general. Cyclic homology (also called “additive K-theory” by Feigin and Tsygan) appears therefore as a first step to compute the K- groups for general algebras. We have tried to make this book as self- contained as possible. Together with the motivations provided by A. Connes, the reader should not find any special difficulty to read it. In particular, the first chapters can be easily integrated in a graduate course on the subject. (Zentralblatt)

Bibliogr. p. 140-142. Index

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