About: Severi–Brauer variety

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In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K. Francesco Severi studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. 1 → GL1 → GLn → PGLn → 1 of algebraic groups. This implies a connecting homomorphism H1(PGLn) → H2(GL1)

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rdfs:label	Severi–Brauer variety (en)
rdfs:comment	In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K. Francesco Severi studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. 1 → GL1 → GLn → PGLn → 1 of algebraic groups. This implies a connecting homomorphism H1(PGLn) → H2(GL1) (en)
dcterms:subject	Algebraic varieties Algebraic groups Ring theory Diophantine geometry Homological algebra
Wikipage page ID	1669368 (xsd:integer)
Wikipage revision ID	1000187430 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Projective space Projective bundle Algebraic closure Algebraic group Algebraic varieties Richard Brauer Algebraic groups Ring theory Conic section Mathematics Short exact sequence Matrix ring Galois cohomology Linear system Algebraic variety American Mathematical Society Field (mathematics) Diophantine geometry Brauer group Central simple algebra Diophantine geometry Hilbert's Theorem 90 Projective line Projective linear group Isomorphic Homological algebra Quaternion algebra Perfect field Springer-Verlag Divisor class Connecting homomorphism Dimension of a variety
Link from a Wikipage to an external page	http://ebooks.cambridge.org/chapter.jsf%3Fbid=CBO9780511607219&cid=CBO9780511607219A036 http://www.mathcs.emory.edu/~brussel/Papers/galoisdescent.pdf
sameAs	Severi–Brauer variety Severi–Brauer variety Severi–Brauer variety
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Reflist dbt:Harvs
authorlink	Francesco Severi (en)
first	Francesco (en)
id	b/b017620 (en)
last	Severi (en)
title	Brauer–Severi variety (en)
year	1932 (xsd:integer)
has abstract	In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K. Francesco Severi studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra (a,b)K corresponds to the conic C(a,b) with equation and the algebra (a,b)K splits, that is, (a,b)K is isomorphic to a matrix algebra over K, if and only if C(a,b) has a point defined over K: this is in turn equivalent to C(a,b) being isomorphic to the projective line over K. 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 inH1(PGLn),where PGLnis the projective linear group, and n is the dimension of the variety V. There is a short exact sequence 1 → GL1 → GLn → PGLn → 1 of algebraic groups. This implies a connecting homomorphism H1(PGLn) → H2(GL1) at the level of cohomology. Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial becauseH1(GLn) = {1}by an extension of Hilbert's Theorem 90. Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras. Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence Here the map δ sends 1 to the Brauer class corresponding to X. As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degree d on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L. (en)
prov:wasDerivedFrom	wikipedia-en:Severi–Brauer_variety?oldid=1000187430&ns=0
page length (characters) of wiki page	6032 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Severi–Brauer_variety
is Link from a Wikipage to another Wikipage of	Projective space Norm variety Rational variety Principal homogeneous space Projective bundle Glossary of ring theory Pseudo algebraically closed field Galois cohomology Francesco Severi Brauer group Central simple algebra Brauer–Severi variety Severi-Brauer variety Severi-Brauer surface Severi–Brauer surface Brauer-Severi variety Brauer variety
is Wikipage redirect of	Brauer–Severi variety Severi-Brauer variety Severi-Brauer surface Severi–Brauer surface Brauer-Severi variety Brauer variety
is foaf:primaryTopic of	wikipedia-en:Severi–Brauer_variety

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