Cyclic Homology pp 337-376 | Cite as

# Algebraic *K*-Theory

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## Abstract

The Grothendieck group *K* _{0}(*A*) of a ring *A* was introduced in the sixties by Grothendieck in order to give a nice formulation of the Riemann-Roch theorem. Then it was recognized that the Grothendieck group is closely related to the abelianization *K* _{1}(*A*) of the general linear group, which had been studied earlier (1949) by J.H.C. Whitehead in his work on simple homotopy. The next step was the discovery of the *K* _{2}-group by Milnor in his attempt to understand the Steinberg symbols in arithmetic. At that point these three groups were expected to be part of a family of algebraic *K*-functors *K* _{ n } defined for all *n* ≥ 0. After several attempts by different people, Quillen came with a simple construction, the so-called plus-construction, which gives rise to higher algebraic *K*-theory.

## Keywords

Exact Sequence Banach Algebra General Linear Group Chern Character Cyclic Homology## Preview

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## Bibliographical Comments on Chapter 11

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