Central Simple Algebras and Galois CohomologyThis 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. |
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Affine and projective affine closed set affine variety Assume moreover automorphism B-module b₁ birational called closed points closed subset closed subvariety codimension coefficients complete discrete valuation coordinate ring Corollary D+(xi defined denote DerA(B derivation differential forms discrete valuation ring div(f Div(X divisor elements exact sequence extension L|K field extension finite extension follows formula fraction field Galois extension Galois group homogeneous polynomials hyperplane integral closure irreducible isomorphism Jacobson polynomials k-rational k-rational points k-vector space k[xo Krull dimension Lemma Lie algebra Lie bracket linear system maximal ideal module of differential morphism nonzero notion p-basis perfect field Picard group prime ideal projective space projective variety Proof See Matsumura Proof See Serre Proposition A.8.3 quotient rational function rational map residue field ring Ox ring with maximal satisfying scheme-theoretic point Section separable Shafarevich smooth Theorem transcendence degree Ud+1 x₁ Y C X Zariski topology