Central Simple Algebras and Galois Cohomology by Philippe Gille

By Philippe Gille

This ebook is the 1st complete, smooth creation to the idea of significant easy algebras over arbitrary fields. ranging from the fundamentals, it reaches such complex effects because the Merkurjev-Suslin theorem. This theorem is either the fruits of labor initiated by means of Brauer, Noether, Hasse and Albert and the start line of present learn in motivic cohomology idea by way of Voevodsky, Suslin, Rost and others. Assuming just a sturdy historical past in algebra, yet no homological algebra, the ebook covers the fundamental conception of significant uncomplicated algebras, tools of Galois descent and Galois cohomology, Severi-Brauer kinds, residue maps and, ultimately, Milnor K-theory and K-cohomology. The final bankruptcy rounds off the speculation via proposing the implications 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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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. 32 Central simple algebras and Galois descent Before proving the proposition, we recall a notion from ring theory: the opposite algebra A◦ of a k-algebra A is the k-algebra with the same underlying k-vector space as A, but in which the product of two elements x, y is given by the element yx with respect to the product in A.

2 Splitting fields The last corollary enables one to give an alternative characterization of central simple algebras. 1 Let k be a field and A a finite dimensional k-algebra. Then A is a central simple algebra if and only if there exist an integer n > 0 and a finite field extension K |k so that A ⊗k K is isomorphic to the matrix ring Mn (K ). 2 Let A be a finite dimensional k-algebra, and K |k a finite field extension. The algebra A is central simple over k if and only if A ⊗k K is central simple over K .

One sees from the definition that each Brauer equivalence class contains (up to isomorphism) a unique division algebra. 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.

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