Milne, J. S. Arithmetic Duality Theorems

ISBN 13: 9781419642746

Arithmetic Duality Theorems

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9781419642746: Arithmetic Duality Theorems


This book, intended for research mathematicians, proves the duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry, for example, in the proof of Fermat's Last Theorem.

Reviews of the first edition
The book deals with duality theorems in Galois, étale and flat cohomology, for local and global fields, as well as the corresponding rings of integers. Also covered are results about cohomological dimension, finiteness and Euler-Poincaré characteristics. It can serve as a good general reference for these questions.
Mathematical Reviews, Gerd Faltings.

However, much of this work [by Tate, Artin, Verdier, and others] was never published in details. The main purpose of the book under review is to offer a selfcontained and systematic treatment of these developments.
Zentralblatt MATH, L. Badescu.

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From the Back Cover:

This volume presents for the first time complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. Chapter 1 is devoted to an exposition of these theorems in the Galois cohomology of number fields announced by Tate in 1962 and describes later work in the same area. The discussion assumes only a knowledge of basic Galois cohomology and class field theory.
Chapter 2 focuses on the work of Artin and Verdier who re-interpreted and developed Tate's ideas in the framework of etale cohomology; some of the more recent developments in this area are also covered.
Finally, in Chapter 3, which contains a number of new results, it is shown how flat cohomology is needed in order to prove and to apply duality theorems in the case of groups which have torsion of order divisible by one of the residue characteristics.

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Milne, J. S.
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