Algebraic K-theory of Number Fields

✍ Scribed by Alexey Beshenov

• Year: 2014
• Tongue: English
• Leaves: 149
• Series: Master thesis at Università degli Studi di Milano / Université de Bordeaux
• Edition: version 1 Jul 2014
• Category: Library

✦ Table of Contents

Preface......Page 3
K0 of a ring......Page 13
K1 of a ring......Page 18
A few words about K2......Page 28
Higher algebraic K-theory of rings (plus-construction)......Page 33
Plus-construction for a space......Page 34
Homotopy groups of X+......Page 38
Higher K-groups of a ring......Page 40
Quillen's results......Page 42
Rational homotopy: from ranks of K-groups to cohomology of SL......Page 47
H-spaces......Page 48
Hopf algebras......Page 49
Rationalization of H-spaces......Page 54
Cartan–Serre theorem......Page 57
The setting......Page 61
De Rham complex......Page 63
Group cohomology......Page 66
Lie algebra cohomology......Page 68
Relative Lie algebra cohomology......Page 72
Cohomology and homotopy of SU/SO(R) and SU......Page 77
The morphism j: I –> H (Gamma, R)......Page 78
Final results......Page 79
Harmonic forms on a compact manifold (théorie de Hodge pour les nuls)......Page 83
Matsushima's constant......Page 88
Matsushima's theorem......Page 92
Complete Riemannian manifolds......Page 97
Adjunction = on complete manifolds......Page 100
Square integrable forms......Page 102
A Stokes' formula for complete Riemannian manifolds......Page 103
Garland's theorem......Page 104
Basic definitions......Page 107
Extension and restriction of scalars......Page 108
Arithmetic groups......Page 109
Hurewicz theorem......Page 111
Fibrations and cofibrations......Page 112
Leray–Serre spectral sequence......Page 114
Acyclic maps......Page 117
K0 of a category......Page 121
Simplicial sets and their geometric realization......Page 122
Classifying space of a category......Page 126
Coverings......Page 127
Exact categories......Page 129
The category QC......Page 131
Higher K-groups via the Q-construction......Page 135
Quotient categories......Page 138
Quillen's results......Page 141