Central Simple Algebras and Galois Cohomology by Philippe Gille

By Philippe Gille

This publication is the 1st finished, smooth creation to the idea of critical easy algebras over arbitrary fields. ranging from the fundamentals, it reaches such complicated effects because the Merkurjev-Suslin theorem. This theorem is either the end result of labor initiated by means of Brauer, Noether, Hasse and Albert and the start line of present learn in motivic cohomology thought by way of Voevodsky, Suslin, Rost and others. Assuming just a stable historical past in algebra, yet no homological algebra, the booklet covers the elemental idea of important basic algebras, tools of Galois descent and Galois cohomology, Severi-Brauer kinds, residue maps and, eventually, Milnor K-theory and K-cohomology. The final bankruptcy rounds off the speculation by way of providing the consequences in optimistic attribute, together with the concept of Bloch-Gabber-Kato. The ebook is acceptable as a textbook for graduate scholars and as a reference for researchers operating in algebra, algebraic geometry or K-theory.

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Thus we can also say that Br (K |k) classifies division algebras split by K . It follows from Wedderburn’s theorem and the previous remark that if A and B are two Brauer equivalent k-algebras of the same dimension, then A∼ = B. The set Br (K |k) (and hence also Br (k)) is equipped with a product operation induced by tensor product of k-algebras; indeed, the tensor product manifestly preserves Brauer equivalence. 8 The sets Br (K |k) and Br (k) equipped with the above product operation are abelian groups.

The main theorem on simple algebras over a field provides a converse to the above example. 3 (Wedderburn) Let A be a finite dimensional simple algebra over a field k. Then there exist an integer n ≥ 1 and a division algebra D ⊃ k so that A is isomorphic to the matrix ring Mn (D). Moreover, the division algebra D is uniquely determined up to isomorphism. The proof will follow from the next two lemmas. Before stating them, let us recall some basic facts from module theory. First, a nonzero A-module M is simple if it has no A-submodules other than 0 and M.

Thus the class of a tensor product of degree m cyclic algebras has order dividing m in the Brauer group. 7 (Merkurjev–Suslin) Assume that k contains a primitive m-th root of unity ω. Then a central simple k-algebra whose class has order dividing m in Br (k) is Brauer equivalent to a tensor product (a1 , b1 )ω ⊗k · · · ⊗k (ai , bi )ω of cyclic algebras. This generalizes Merkurjev’s theorem from the end of Chapter 1. In fact, Merkurjev and Suslin found this generalization soon after the first result of Merkurjev.

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