Severi-Brauer variety

In mathematics, a Severi-Brauer variety over a field K is an algebraic variety V which becomes isomorphic to projective space over an algebraic closure of K. Examples are conic sections C: provided C is non-singular, it becomes isomorphic to the projective line over any extension field L over which C has a point defined. The name is for Francesco Severi and Richard Brauer.

Related Topics:
Mathematics - Field - Algebraic variety - Isomorphic - Projective space - Algebraic closure - Conic section - Non-singular - Projective line - Extension field - Francesco Severi - Richard Brauer

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Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (at least if K is a perfect field) Galois cohomology classes in

Related Topics:
Diophantine geometry - Galois cohomology - Perfect field

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:H1(PGLn)

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in the projective linear group, where n is the dimension of V. There is a short exact sequence

Related Topics:
Projective linear group - Dimension - Short exact sequence

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:1 → GL1 → GLn → PGLn → 1

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of algebraic groups. This implies a connecting homomorphism

Related Topics:
Algebraic group - Connecting homomorphism

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:H1(PGLn) → H2(GL1)

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at the level of cohomology. The RHS is identified with the Brauer group of K, while the kernel is trivial because

Related Topics:
RHS - Brauer group

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:H1(GLn) = {1}

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by an extension of Hilbert's Theorem 90. Therefore the Severi-Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras.

Related Topics:
Hilbert's Theorem 90 - Central simple algebra

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