Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics, Series Number 117)

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. 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 "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." -- Publisher's description This book presents the connection between Galois groups in algebra and fundamental groups in topology. Starting at an elementary level, it shows 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