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E-grāmata: Galois Groups and Fundamental Groups

3.71/5 (14 ratings by Goodreads)
(Hungarian Academy of Sciences, Budapest)
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Galois Groups and Fundamental GroupsPalielināt
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Assuming little technical background, the author presents the strong analogies between these two concepts starting at an elementary level.

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.

Recenzijas

"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

Papildus informācija

Assuming little technical background, the author presents the strong analogies between these two concepts starting at an elementary level.
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 etale 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 etale covers of schemes
152
5.3 Galois theory for finite etale 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
Tamįs Szamuely is a Senior Research Fellow in the Alfréd Rényi Institute of Mathematics at the Hungarian Academy of Sciences, Budapest.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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