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Galois cohomology and class field theory / David Harari.

By: Contributor(s): Material type: TextTextLanguage: English Original language: French Series: UniversitextPublication details: Cham : Springer, 2020.Description: 1 online resource (336 p.)ISBN:
  • 9783030439019
  • 3030439011
Uniform titles:
  • Cohomologie galoisienne et théorie du corps de classes. English.
Subject(s): DDC classification:
  • 514.23 23 HAR
LOC classification:
  • QA612.3
Online resources:
Contents:
Intro -- Preface -- Notation and Conventions -- Contents -- Part I Group Cohomology and Galois Cohomology: Generalities -- 1 Cohomology of Finite Groups: Basic Properties -- 1.1 The Notion of G-Module -- 1.2 The Category of G-Modules -- 1.3 The Cohomology Groups HGA -- 1.4 Computation of Cohomology Using the Cochains -- 1.5 Change of Group: Restriction, Corestriction, the Hochschild-Serre ... -- 1.6 Corestriction -- Applications -- 1.7 Exercises -- 2 Groups Modified à la Tate, Cohomology of Cyclic Groups -- 2.1 Tate Modified Cohomology Groups. -- 2.2 Change of Group. Transfer.
2.3 Cohomology of a Cyclic Group -- 2.4 Herbrand Quotient -- 2.5 Cup-Products -- 2.6 Cup-Products for the Modified Cohomology -- 2.7 Exercises -- 3 p-Groups, the Tate-Nakayama Theorem -- 3.1 Cohomologically Trivial Modules -- 3.2 The Tate-Nakayama Theorem -- 3.3 Exercises -- 4 Cohomology of Profinite Groups -- 4.1 Basic Facts About Profinite Groups -- 4.2 G-Modules -- 4.3 Cohomology of a Discrete G-Module -- 4.4 Exercises -- 5 Cohomological Dimension -- 5.1 Definitions, First Examples -- 5.2 Properties of the Cohomological Dimension -- 5.3 Exercises -- 6 First Notions of Galois Cohomology.
6.1 Generalities -- 6.2 Hilbert's Theorem 90 and Applications -- 6.3 Brauer Group of a Field -- 6.4 Cohomological Dimension of a Field -- 6.5 C1 -- 6.6 Exercises -- Part II Local Fields -- 7 Basic Facts About Local Fields -- 7.1 Discrete Valuation Rings -- 7.2 Complete Field for a Discrete Valuation -- 7.3 Extensions of Complete Fields -- 7.4 Galois Theory of a Complete Field for a Discrete Valuation -- 7.5 Structure Theorem -- Filtration of the Group of Units -- 7.6 Exercises -- 8 Brauer Group of a Local Field -- 8.1 Local Class Field Axiom -- 8.2 Computation of the Brauer Group.
8.3 Cohomological Dimension -- Finiteness Theorem -- 8.4 Exercises -- 9 Local Class Field Theory: The Reciprocity Map -- 9.1 Definition and Main Properties -- 9.2 The Existence Theorem: Preliminary Lemmas and the Case of a p-adic Field -- 9.3 Exercises -- 10 The Tate Local Duality Theorem -- 10.1 The Dualising Module -- 10.2 The Local Duality Theorem -- 10.3 The Euler-Poincaré Characteristic -- 10.4 Unramified Cohomology -- 10.5 From the Duality Theorem to the Existence Theorem -- 10.6 Exercises -- 11 Local Class Field Theory: Lubin-Tate Theory -- 11.1 Formal Groups.
11.2 Change of the Uniformiser -- 11.3 Fields Associated to Torsion Points -- 11.4 Computation of the Reciprocity Map -- 11.5 The Existence Theorem (the General Case) -- 11.6 Exercises -- Part III Global Fields -- 12 Basic Facts About Global Fields -- 12.1 Definitions, First Properties -- 12.2 Galois Extensions of a Global Field -- 12.3 Idèles, Strong Approximation Theorem -- 12.4 Some Complements in the Case of a Function Field -- 12.5 Exercises -- 13 Cohomology of the Idèles: The Class Field Axiom -- 13.1 Cohomology of the Idèle Group -- 13.2 The Second Inequality -- 13.3 Kummer Extensions.
Summary: 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 Čebotarev 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.
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Holdings
Item type Current library Call number Status Date due Barcode
Project book Project book CUTN Central Library Sciences 514.23 HAR (Browse shelf(Opens below)) Checked out to Renuka Devi V (20019T) 31/01/2024 48831

Description based upon print version of record.

13.4 First Inequality and the Axiom of Class Field Theory.

Includes bibliographical references and index.

Intro -- Preface -- Notation and Conventions -- Contents -- Part I Group Cohomology and Galois Cohomology: Generalities -- 1 Cohomology of Finite Groups: Basic Properties -- 1.1 The Notion of G-Module -- 1.2 The Category of G-Modules -- 1.3 The Cohomology Groups HGA -- 1.4 Computation of Cohomology Using the Cochains -- 1.5 Change of Group: Restriction, Corestriction, the Hochschild-Serre ... -- 1.6 Corestriction -- Applications -- 1.7 Exercises -- 2 Groups Modified à la Tate, Cohomology of Cyclic Groups -- 2.1 Tate Modified Cohomology Groups. -- 2.2 Change of Group. Transfer.

2.3 Cohomology of a Cyclic Group -- 2.4 Herbrand Quotient -- 2.5 Cup-Products -- 2.6 Cup-Products for the Modified Cohomology -- 2.7 Exercises -- 3 p-Groups, the Tate-Nakayama Theorem -- 3.1 Cohomologically Trivial Modules -- 3.2 The Tate-Nakayama Theorem -- 3.3 Exercises -- 4 Cohomology of Profinite Groups -- 4.1 Basic Facts About Profinite Groups -- 4.2 G-Modules -- 4.3 Cohomology of a Discrete G-Module -- 4.4 Exercises -- 5 Cohomological Dimension -- 5.1 Definitions, First Examples -- 5.2 Properties of the Cohomological Dimension -- 5.3 Exercises -- 6 First Notions of Galois Cohomology.

6.1 Generalities -- 6.2 Hilbert's Theorem 90 and Applications -- 6.3 Brauer Group of a Field -- 6.4 Cohomological Dimension of a Field -- 6.5 C1 -- 6.6 Exercises -- Part II Local Fields -- 7 Basic Facts About Local Fields -- 7.1 Discrete Valuation Rings -- 7.2 Complete Field for a Discrete Valuation -- 7.3 Extensions of Complete Fields -- 7.4 Galois Theory of a Complete Field for a Discrete Valuation -- 7.5 Structure Theorem -- Filtration of the Group of Units -- 7.6 Exercises -- 8 Brauer Group of a Local Field -- 8.1 Local Class Field Axiom -- 8.2 Computation of the Brauer Group.

8.3 Cohomological Dimension -- Finiteness Theorem -- 8.4 Exercises -- 9 Local Class Field Theory: The Reciprocity Map -- 9.1 Definition and Main Properties -- 9.2 The Existence Theorem: Preliminary Lemmas and the Case of a p-adic Field -- 9.3 Exercises -- 10 The Tate Local Duality Theorem -- 10.1 The Dualising Module -- 10.2 The Local Duality Theorem -- 10.3 The Euler-Poincaré Characteristic -- 10.4 Unramified Cohomology -- 10.5 From the Duality Theorem to the Existence Theorem -- 10.6 Exercises -- 11 Local Class Field Theory: Lubin-Tate Theory -- 11.1 Formal Groups.

11.2 Change of the Uniformiser -- 11.3 Fields Associated to Torsion Points -- 11.4 Computation of the Reciprocity Map -- 11.5 The Existence Theorem (the General Case) -- 11.6 Exercises -- Part III Global Fields -- 12 Basic Facts About Global Fields -- 12.1 Definitions, First Properties -- 12.2 Galois Extensions of a Global Field -- 12.3 Idèles, Strong Approximation Theorem -- 12.4 Some Complements in the Case of a Function Field -- 12.5 Exercises -- 13 Cohomology of the Idèles: The Class Field Axiom -- 13.1 Cohomology of the Idèle Group -- 13.2 The Second Inequality -- 13.3 Kummer Extensions.

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 Čebotarev 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.

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