Author(s): Tamas Szamuely
Series: Cambridge Studies in Advanced Mathematics 117
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 282
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Acknowledgments......Page 11
1.1 Algebraic field extensions......Page 13
1.2 Galois extensions......Page 16
1.3 Infinite Galois extensions......Page 21
1.4 Interlude on category theory......Page 27
1.5 Finite etale algebras......Page 32
Exercises......Page 37
2.1 Covers......Page 39
2.2 Galois covers......Page 42
2.3 The monodromy action......Page 46
2.4 The universal cover......Page 51
2.5 Locally constant sheaves and their classification......Page 57
2.6 Local systems......Page 63
2.7 The Riemann-Hilbert correspondence......Page 66
Exercises......Page 74
3.1 Basic concepts......Page 77
3.2 Local structure of holomorphic maps......Page 79
3.3 Relation with field theory......Page 84
3.4 The absolute Galois group of C(t)......Page 90
3.5 An alternate approach: patching Galois covers......Page 95
3.6 Topology of Riemann surfaces......Page 98
Exercises......Page 103
4.1 Background in commutative algebra......Page 105
4.2 Curves over an algebraically closed field......Page 111
4.3 Affine curves over a general base field......Page 117
4.4 Proper normal curves......Page 122
4.5 Finite branched covers of normal curves......Page 126
4.6 The algebraic fundamental group......Page 131
4.7 The outer Galois action......Page 135
4.8 Application to the inverse Galois problem......Page 141
4.9 A survey of advanced results......Page 146
Exercises......Page 152
5.1 The vocabulary of schemes......Page 154
5.2 Finite étale covers of schemes......Page 164
5.3 Galois theory for finite étale covers......Page 171
5.4 The algebraic fundamental group in the general case......Page 178
5.5 First properties of the fundamental group......Page 182
5.6 The homotopy exact sequence and applications......Page 187
5.7 Structure theorems for the fundamental group......Page 194
5.8 The abelianized fundamental group......Page 205
Exercises......Page 215
6.1 Affine group schemes and Hopf algebras......Page 218
6.2 Categories of comodules......Page 226
6.3 Tensor categories and the Tannaka–Krein theorem......Page 234
6.4 Second interlude on category theory......Page 240
6.5 Neutral Tannakian categories......Page 244
6.6 Differential Galois groups......Page 254
6.7 Nori's fundamental group scheme......Page 260
Exercises......Page 271
Bibliography......Page 273
Index......Page 280
