Higher algebraic \(K\)-theory: an overview
type: Item
instance of: scholarly article
title: Higher algebraic \(K\)-theory: an overview
zbMATH Open document ID: 0746.19001
author: Emilio Lluis-Puebla
author: J.-L. Loday
author: Henri Gillet
author: Christophe Soulé
author: Victor P. Snaith
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: This book consists of five papers based on lectures delivered in 1985. It is intended as an introduction to the literature of higher algebraic \(K\)- theory; proofs are often omitted. The five papers are: Introduction to algebraic \(K\)-theory (\textit{Emilio Lluis-Puebla}), Introduction to algebraic \(K\)-theory and cyclic homology (\textit{Jean-Louis Loday}), Comparing algebraic and topological \(K\)-theory (\textit{Henri Gillet}), Algebraic \(K\)-theory of the integers (\textit{Christopher Soulé}), Applications of group cohomology to bilinear forms (\textit{Victor Snaith}). The first paper covers the homology of groups and the low-dimensional \(K\)-theory of rings. The second paper covers the algebraic \(K\)-theory and cyclic homology of rings; a more comprehensive version will appear in \textit{J.-L. Loday's} book: Cyclic homology (Springer Verlag, Grundlehren der mathematischen Wissenschaften) (to appear). The third part covers topological \(K\)-theory and the algebraic \(K\)-theory of varieties and schemes. It concludes with a proof of Suslin's rigidity theorem. The fourth part is mainly devoted to connections with the Bernoulli numbers. The fifth part contains material on group cohomology, Galois cohomology and the Hasse-Witt characteristic classes of bilinear forms. There are new proofs of results of Serre and Fröhlich. There is also a summary of work of \textit{V. Snaith} [Topology 27, 119-127 (1988; Zbl 0696.12012)].
Mathematics Subject Classification ID: 19-02
Mathematics Subject Classification ID: 19Dxx
zbMATH DE Number: 52853
zbMATH Keywords: introductory survey
zbMATH Keywords: higher algebraic \(K\)-theory
zbMATH Keywords: homology of groups
zbMATH Keywords: low-dimensional \(K\)-theory of rings
zbMATH Keywords: cyclic homology of rings
zbMATH Keywords: topological \(K\)-theory
zbMATH Keywords: algebraic \(K\)-theory of varieties and schemes
zbMATH Keywords: Suslin's rigidity theorem
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: group cohomology
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Hasse-Witt characteristic classes of bilinear forms
reviewed by: Richard John Steiner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0088876
OpenAlex ID: W2489514946
DOI: 10.1007/BFB0088876
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description: scientific article; zbMATH DE number 52853