文档介绍：Algebraic K-theory, periodic cyclic
homology, and the Connes-Moscovici
Index Theorem
Master’s thesis by
Bram Mesland
under supervision of
prof dr. . Landsman
University of Amsterdam, Faculty of Science
Korteweg-de Vries Institute for Mathematics
Plantage Muidergracht 24, 1018 TV Amsterdam
June 20th, 2005
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Does it matter that this waste of time is
what makes a life for you?
-Frank Zappa
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Algebraic K-theory, cyclic homology, and the Connes-Moscovici
Index Theorem
Abstract. We develop algebraic K-theory and cyclic homology from
scratch. The boundary map in periodic cyclic cohomology is shown to
be well-behaved with respect to the external product. Then we prove
that the Chern-Connes character induces a natural transformation from
the exact sequence in lower algebraic K-theory to the exact sequence in
periodic cyclic homology. Using this, the Gohberg-Krein index theorem
is easily derived. Finally, we prove the Connes-Moscovici index theorem,
closely following Nistor in [20].
Keywords: Algebraic K-theory, cyclic homology, Chern-Connes char-
acter, index theorem, mutative geometry.
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Contents
Introduction 9
1 Lower algebraic K-theory 15
Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Grothendieck’s K0 . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Whitehead’s K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Relative K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Topological K-theory . . . . . . . . . . . . . . . . . . . . . . . . . 33
C∗-algebras and index theorems . . . . . . . . . . . . . . . . . . . 38
2 Cyclic homology 41
The simplicial and cyclic categories . . . . . . . . . . . . . . . . . 41
Cyclic modules . . . . . . . . . . . . . . . . . . .