Central Simple Algebras and Galois Cohomology (Cambridge Studies in Advanced Mathematics, Series Number 101)

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. 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. Quaternion Algebras -- Central Simple Algebras And Galois Descent -- Techniques From Group Cohomology -- The Cohomological Brauer Group -- Severi-brauer Varieties -- Residue Maps -- Milnor K-theory -- The Merkurjev-suslin Theorem -- Symbols In Positive Characteristic -- A Breviary Of Algebraic Geometry. Philippe Gille, Tamás Szamuely. Includes Bibliographical References (p. [323]-338) And Index. The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Assuming only a solid background in algebra, it reaches such advanced results as the Merkurjev-Suslin theorem. The book is a graduate textbook and a reference for researchers working in algebra, algebraic geometry or K-theory