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AN INTRODUCTION TO ALGEBRAIC K-THEORY Christian Ausoni
 

Summary: AN INTRODUCTION TO ALGEBRAIC K-THEORY
Christian Ausoni
Abstract. These are the notes of an introductory lecture given at The 20th Winter
School for Geometry and Physics, at Srni. It was meant as a leisurely exposition of
classical aspects of algebraic K-theory, with some of its applications to geometry and
topology.
Introduction
Classically, algebraic K-theory of rings is the study of the family of K-theory
functors
Kn : Rings - Abelian Groups (n = 0, 1, 2).
For a given ring R, the groups K0R, K1R and K2R were defined, around the 60's,
in purely algebraic terms, and are closely related to classical invariants of rings. It
soon became apparent that these functors were part of a kind of homology theory
for rings, but no algebraic definition of higher K-groups K3, K4, . . . was found.
In the early 70's, D. Quillen came up with a definition that requires the use of
homotopy theory. He defined the group KnR as the n-th homotopy group of a
certain algebraic K-theory space KR :
KnR = n(KR) (n = 0, 1, 2, . . . ).
Although its construction is quite obscure, the space KR has very nice properties,
and Quillen's definition of KnR, which agrees with the classical one if n = 0, 1, 2,

  

Source: Ausoni, Christian - Institut für Mathematische Statistik, Westfälische Wilhelms Universität Münster

 

Collections: Mathematics