Cover image for Central simple algebras and galois cohomology
Title:
Central simple algebras and galois cohomology
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Series:
Cambridge studies in advanced mathematics ; 101
Publication Information:
New York : Cambridge University Press, 2006
Physical Description:
ix, 343 p. : ill. ; 24 cm.
ISBN:
9780521861038
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30000010229557 QA612.3 G54 2006 Open Access Book
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Summary

Summary

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.


Table of Contents

1 Quaternion algebras
2 Central simple algebras and Galois descent
3 Techniques from group cohomology
4 The cohomological Brauer group
5 Severi-Brauer varieties
6 Residue maps
7 Milnor K-theory
8 The Merkurjev-Suslin theorem
9 Symbols in positive characteristic
Appendix: A breviary of algebraic geometry
References
Index