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Galois Cohomology and Class Field Theory 2020 ed. [Minkštas viršelis]

Translated by ,
  • Formatas: Paperback / softback, 338 pages, aukštis x plotis: 235x155 mm, weight: 539 g, 2 Illustrations, color; 48 Illustrations, black and white; XIV, 338 p. 50 illus., 2 illus. in color., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 24-Jun-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030439003
  • ISBN-13: 9783030439002
Kitos knygos pagal šią temą:
Galois Cohomology and Class Field Theory 2020 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 338 pages, aukštis x plotis: 235x155 mm, weight: 539 g, 2 Illustrations, color; 48 Illustrations, black and white; XIV, 338 p. 50 illus., 2 illus. in color., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 24-Jun-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030439003
  • ISBN-13: 9783030439002
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030439002.html
This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory.

Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Cebotarev density theorem.

Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

Recenzijos

This book is a very good textbook for studying important roles of Galois cohomology in algebraic number theory . This book consists of four parts, and the chapters are well constructed. Examples and exercises are also well selected and interesting. This English translation is also very useful not only for students but also for researchers who study algebraic number theory in modern mathematical language, such as that of category theory or homological algebra. (Yasushi Mizusawa, Mathematical Reviews, April, 2022) For students with no prior understanding of class field theory, the book is ideal. It is self-contained, and is based on a concatenation of masters level courses given by the author. his book seamlessly stitches together all the components in a neat and lucid manner. The whole process of learning this classical theory from Hararis book makes it a painless and enjoyable experience. The author takes a lot of care to make illuminating remarks in each chapter (Balasubramanian Sury, zbMATH 1466.11086, 2021)

Part I Group Cohomology and Galois Cohomology: Generalities
1 Cohomology of Finite Groups: Basic Properties
3(26)
1.1 The Notion of G-Module
4(2)
1.2 The Category of G-Modules
6(4)
1.3 The Cohomology Groups Hi(G,A)
10(4)
1.4 Computation of Cohomology Using the Cochains
14(4)
1.5 Change of Group: Restriction, Corestriction, the Hochschild-Serre Spectral Sequence
18(6)
1.6 Corestriction; Applications
24(2)
1.7 Exercises
26(3)
2 Groups Modified a la Tate, Cohomology of Cyclic Groups
29(26)
2.1 Tate Modified Cohomology Groups.
29(4)
2.2 Change of Group. Transfer
33(7)
2.3 Cohomology of a Cyclic Group
40(1)
2.4 Herbrand Quotient
41(1)
2.5 Cup-Products
42(3)
2.6 Cup-Products for the Modified Cohomology
45(8)
2.7 Exercises
53(2)
3 p-Groups, the Tate-Nakayama Theorem
55(10)
3.1 Cohomologically Trivial Modules
55(5)
3.2 The Tate-Nakayama Theorem
60(3)
3.3 Exercises
63(2)
4 Cohomology of Profinite Groups
65(14)
4.1 Basic Facts About Profinite Groups
65(5)
4.2 Discrete G-Modules
70(1)
4.3 Cohomology of a Discrete G-Module
71(5)
4.4 Exercises
76(3)
5 Cohomological Dimension
79(8)
5.1 Definitions, First Examples
79(2)
5.2 Properties of the Cohomological Dimension
81(3)
5.3 Exercises
84(3)
6 First Notions of Galois Cohomology
87(12)
6.1 Generalities
87(1)
6.2 Hilbert's Theorem 90 and Applications
88(1)
6.3 Brauer Group of a Field
89(2)
6.4 Cohomological Dimension of a Field
91(1)
6.5 C1 Fields
92(2)
6.6 Exercises
94(5)
Part II Local Fields
7 Basic Facts About Local Fields
99(10)
7.1 Discrete Valuation Rings
99(1)
7.2 Complete Field for a Discrete Valuation
100(1)
7.3 Extensions of Complete Fields
101(2)
7.4 Galois Theory of a Complete Field for a Discrete Valuation
103(1)
7.5 Structure Theorem; Filtration of the Group of Units
104(1)
7.6 Exercises
105(4)
8 Brauer Group of a Local Field
109(10)
8.1 Local Class Field Axiom
109(1)
8.2 Computation of the Brauer Group
110(4)
8.3 Cohomological Dimension; Finiteness Theorem
114(1)
8.4 Exercises
115(4)
9 Local Class Field Theory: The Reciprocity Map
119(12)
9.1 Definition and Main Properties
119(5)
9.2 The Existence Theorem: Preliminary Lemmas and the Case of a p-adic Field
124(5)
9.3 Exercises
129(2)
10 The Tate Local Duality Theorem
131(14)
10.1 The Dualising Module
131(4)
10.2 The Local Duality Theorem
135(2)
10.3 The Euler-Poincare Characteristic
137(1)
10.4 Unramified Cohomology
138(2)
10.5 From the Duality Theorem to the Existence Theorem
140(1)
10.6 Exercises
141(4)
11 Local Class Field Theory: Lubin-Tate Theory
145(16)
11.1 Formal Groups
145(4)
11.2 Change of the Uniformiser
149(3)
11.3 Fields Associated to Torsion Points
152(1)
11.4 Computation of the Reciprocity Map
153(3)
11.5 The Existence Theorem (the General Case)
156(2)
11.6 Exercises
158(3)
Part III Global Fields
12 Basic Facts About Global Fields
161(16)
12.1 Definitions, First Properties
161(3)
12.2 Galois Extensions of a Global Field
164(1)
12.3 Ideles, Strong Approximation Theorem
165(6)
12.4 Some Complements in the Case of a Function Field
171(2)
12.5 Exercises
173(4)
13 Cohomology of the Ideles: The Class Field Axiom
177(20)
13.1 Cohomology of the Idele Group
177(4)
13.2 The Second Inequality
181(4)
13.3 Kummer Extensions
185(2)
13.4 First Inequality and the Axiom of Class Field Theory
187(5)
13.5 Proof of the Class Field Axiom for a Function Field
192(4)
13.6 Exercises
196(1)
14 Reciprocity Law and the Brauer-Hasse-Noether Theorem
197(10)
14.1 Existence of a Neutralising Cyclic Extension
197(3)
14.2 The Global Invariant and the Norm Residue Symbol
200(5)
14.3 Exercises
205(2)
15 The Abelianised Absolute Galois Group of a Global Field
207(28)
15.1 Reciprocity Map and the Idele Class Group
207(4)
15.2 The Global Existence Theorem
211(5)
15.3 The Case of a Function Field
216(3)
15.4 Ray Class Fields; Hilbert Class Field
219(5)
15.5 Galois Groups with Restricted Ramification
224(4)
15.6 Exercises
228(7)
Part IV Duality Theorems
16 Class Formations
235(24)
16.1 Notion of Class Formation
235(3)
16.2 The Spectral Sequence of the Ext
238(8)
16.3 The Duality Theorem for a Class Formation
246(4)
16.4 P-Class Formation
250(1)
16.5 From the Existence Theorem to the Duality Theorem for a p-adic Field
251(2)
16.6 Complements
253(3)
16.7 Exercises
256(3)
17 Poitou-Tate Duality
259(20)
17.1 The P-Class Formation Associated to a Galois Group with Restricted Ramification
259(2)
17.2 The Groups PiS(M)
261(3)
17.3 Statement of the Poitou-Tate Theorems
264(2)
17.4 Proof of Poitou-Tate Theorems (I): Computation of the Ext Groups
266(3)
17.5 Proof of the Poitou-Tate Theorems (II): Computation of the Ext with Values in Is and End of the Proof
269(4)
17.6 Exercises
273(6)
18 Some Applications
279(12)
18.1 Triviality of Some of the IIIi
279(6)
18.2 The Strict Cohomological Dimension of a Number Field
285(2)
18.3 Exercises
287(4)
Appendix A: Some Results from Homological Algebra 291(22)
Appendix B: A Survey of Analytic Methods 313(20)
References 333(4)
Index 337
David Harari is a professor at the Université Paris-Sud (Orsay). He is a specialist in arithmetic and algebraic geometry, author of 40 research papers in these fields.