Galois Groups and Fundamental Groups

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Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.

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Editorial Reviews

From the Publisher
"The book is well written and contains much information about the etale fundamental group. There are exercises in every chapter. On the whole, the book is useful for mathematicians and graduate students looking for one place where they can find information about the etale fundamental group and the related Nori fundamental group scheme."
Swaminathan Subramanian, Mathematical Reviews
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Product Details

Meet the Author

Tamás Szamuely is a Senior Research Fellow in the Alfréd Rényi Institute of Mathematics at the Hungarian Academy of Sciences, Budapest.

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Table of Contents

Preface vii

1 Galois theory of fields 1

1.1 Algebraic field extensions 1

1.2 Galois extensions 4

1.3 Infinite Galois extensions 9

1.4 Interlude on category theory 15

1.5 Finite ?tale algebras 20

2 Fundamental groups in topology 27

2.1 Covers 27

2.2 Galois covers 30

2.3 The monodromy action 34

2.4 The universal cover 39

2.5 Locally constant sheaves and their classification 45

2.6 Local systems 51

2.7 The Riemann-Hilbert correspondence 54

3 Riemann surfaces 65

3.1 Basic concepts 65

3.2 Local structure of holomorphic maps 67

3.3 Relation with field theory 72

3.4 The absolute Galois group of C(t) 78

3.5 An alternate approach: patching Galois covers 83

3.6 Topology of Riemann surfaces 86

4 Fundamental groups of algebraic curves 93

4.1 Background in commutative algebra 93

4.2 Curves over an algebraically closed field 99

4.3 Affine curves over a general base field 105

4.4 Proper normal curves 110

4.5 Finite branched covers of normal curves 114

4.6 The algebraic fundamental group 119

4.7 The outer Galois action 123

4.8 Application to the inverse Galois problem 129

4.9 A survey of advanced results 134

5 Fundamental groups of schemes 142

5.1 The vocabulary of schemes 142

5.2 Finite ?tale covers of schemes 152

5.3 Galois theory for finite ?tale covers 159

5.4 The algebraic fundamental group in the general case 166

5.5 First properties of the fundamental group 170

5.6 The homotopy exact sequence and applications 175

5.7 Structure theorems for the fundamental group 182

5.8 The abelianized fundamental group 193

6 Tannakian fundamental groups 206

6.1 Affine group schemes and Hopf algebras 206

6.2 Categories of comodules 214

6.3 Tensor categories and the Tannaka-Krein theorem 222

6.4 Second interlude on category theory 228

6.5 Neutral Tannakian categories 232

6.6 Differential Galois groups 242

6.7 Nori's fundamental group scheme 248

Bibliography 261

Index 268

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