Central Simple Algebras and Galois Cohomology 中心单代数与伽罗瓦上同调

内容概要

Philippe Gille is Chargé de Recherches, CNRS, Université de Paris-Sud, Orsay. Tamás Szamuely is Senior Research Fellow, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest.

书籍目录

Preface

Acknowledgments

1  Quaternion algebras

　1.1  Basic properties

　1.2  Splitting over a quadratic extension

　1.3  The associated conic

　1.4  A theorem of Witt

　1.5  Tensor products ofquaternion algebras

2  Central simple algebras and Galois descent

　2.1  Wedderburn's theorem

　2.2  Splitting fields

　2.3  Galois descent

　2.4  The Brauer group

　2.5  Cyclic algebras

　2.6  Reduced norms and traces

　2.7  A basic exact sequence

　2.8  K1 of central simple algebras

3  Techniques from group cohomology

　3.1  Definition ofcohomology groups

　3.2  Explicit resolutions

　3.3  Relation to subgroups

　3.4  Cup-products

4   The eohomological Brauer group

　4.1  Profinite groups and Galois groups

　4.2  Cohomology ofprofinite groups

　4.3  The cohomology exact sequence　

　4.4  The Brauer group revisited

　4.5  Index and period

　4.6  The Galois symbol

　4.7  Cyclic algebras and symbols

5  Severi-Brauer varieties

　5.1  Basic properties

　5.2  Classification by Galois cohomology

　5.3  Geometric Brauer equivalence

　5.4  Amitsur's theorem

　5.5  An application: making central simple algebras cyclic

6  Residue maps

　6.1   Cohomological dimension

　6.2  Cl-fields

　6.3  Cohomology of Laurent series fields

　6.4  Cohomology of function fields of curves

　6.5  Application to class field theory

　6.6  Application to the rationality problem: the method

　6.7  Application to the rationality problem: the example

　6.8  Residue maps with finite coefficients

　6.9  The Faddeev sequence with finite coefficients

7  Milnor K-theory

　7.1  The tame symbol

　7.2  Milnor's exact sequence and the Bass-Tate lemma

　7.3  The norm map

　7.4  Reciprocity laws

　7.5  Applications to the Galois symbol

　7.6  The Galois symbol over number fields

8  The Merkurjev-Suslin theorem

　8.1   Gersten complexes in Milnor K-theory

　8.2  Properties of Gersten complexes

　8.3  Aproperty ofSeveri Brauer varieties

　8.4  Hilbert's Theorem 90 for K2

　8.5  The Merkurjev Suslin theorem: a special case

　8.6  The Merkurjev-Suslin theorem: the general case

9  Symbols in positive characteristic

　9.1  The theorems of Teichmtiller and Albert

　9.2  Differential forms and p-torsion in the Brauer group

　9.3  Logarithmic differentials and flat p-connections

　9.4  Decomposition of the de Rham complex

　9.5  The Bloch-Gabber-Kato theorem: statement and reductions

　9.6  Surjectivity of the differential symbol

　9.7  Injectivity of the differential symbol

……

Appendix: A breviary of algebraic geometry

Bibliography

Index

作者简介

This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. 

　　Assuming only a solid background in algebra, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. The book is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory. 

    作者简介：

    Philippe Gille is Chargé de Recherches, CNRS, Université de Paris-Sud, Orsay. Tamás Szamuely is Senior Research Fellow, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest.

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