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An Inroduction to K-theory

Eric M. Friedlander∗

Department of Mathematics, Northwestern University, Evanston, USA

Lectures given at the School on Algebraic K-theory and its Applications Trieste, 14 - 25 May 2007

LNS0823001

∗eric@math.northwestern.edu

Contents

0 Introduction 5

1 K0(−), K1(−), and K2(−)    7

1.1 Algebraic K0 of rings    7

1.2 Topological K0  9

1.3 Quasi-projective Varieti . 10

1.4 Algebraic vector bundles . . . . .. . 12

1.5 Examples of Algebraic Vector Bundles. 13

1.6 Picard Group Pic(X) . . . . . . . 14

1.7 K0 of Quasi-projective Varieties . . . 15

1.8 K1 of rings . . . . . .. . . 16

1.9 K2 of rings . . . . . . . . . . . . . 17

2 Classifying spaces and higher K-theory 19

2.1 Recollections of homotopy theory . . . . . . . . 19

2.2 BG . . . . . .  . . . . . 20

2.3 Quillen’s plus construction .  . . 22

2.4 Abelian and exact categories . . .. . 23

2.5 The S−1S construction . .  . 24

2.6 Simplicial sets and the Nerve of a Category . . . . 26

2.7 Quillen’s Q-constructio . . . 28

3 Topological K-theory 29

3.1 The Classifying space BU ×Z .  . 29

3.2 Bott periodicity . . . 32

3.3 Spectra and Generalized Cohomology Theories . . . . . . . . 33

3.4 Skeleta and Postnikov towers .  . . 36

3.5 The Atiyah-Hirzebruch Spectral sequence . . . . 37

3.6 K-theory Operations . . . .. 39

3.7 Applications . . . . . . . . . . . . . . . . . 41

4 Algebraic K-theory and Algebraic Geometry 42 4.1 Schemes . . . . . . . . 42

4.2 Algebraic cycles . . . . .  . . . 44

4.3 Chow Groups . . . . . . . . 46

4.4 Smooth Varieties . . . . . . . .  49

4.5 Chern classes and Chern character . . .. . . . 51

4.6 Riemann-Roch . . . . . . . . . . .  . . . . . . . 53

5 Some Dicult Problems 55

5.1 K∗(Z) . . . . . . . . . 55

5.2 Bass Finiteness Conjecture . . . . . 57

5.3 Milnor K-theory . . .  . 58

5.4 Negative K-groups . . . . . . . . . 59

5.5 Algebraic versus topological vector bundles . . . .  . 60

5.6 K-theory with nite coecients . .. 60

5.7 Etale K-t. . . . 62

5.8 Integral conjectures . . . . . 63

5.9 K-theory and Quadratic Forms . . . . . . . . . 65

6 Beilinson’s vision partially fullled 65 6.1 Motivation . . . . . . 65

6.2 Statement of conjectur. . 66

6.3 Status of Conjectures . . . 67

6.4 The Meaning of the Conjectures . . . 69

6.5 Etale cohomology . . . .  . . . . 71

6.Voevodsky’s sites . . . . . . . . . . . . . . . . .. . . . 74

References 75

An Introduction to K-theory 5

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