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| In [[mathematics]], a '''Severi–Brauer variety''' over a [[field (mathematics)|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 algebra]]s in such a way that the algebra splits over ''K'' if and only if the variety has a point rational over ''K''. {{harvs|txt|first=Francesco |last=Severi|authorlink=Francesco Severi|year=1932}} studied these varieties, and they are also named after [[Richard Brauer]] because of their close relation to the [[Brauer group]].
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| In dimension one, the Severi–Brauer varieties are [[conic section|conics]]. The corresponding central simple algebras are the [[quaternion algebra]]s. The algebra (''a'',''b'')<sub>''K''</sub> corresponds to the conic ''C''(''a'',''b'') with equation
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| :<math>z^2 = ax^2 + by^2 \ </math>
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| and the algebra (''a'',''b'')<sub>''K''</sub> ''splits'', that is, (''a'',''b'')<sub>''K''</sub> is isomorphic to a [[Matrix ring|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''.<ref name=GS129>Gille & Szamuely (2006) p.129</ref>
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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
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| :''H''<sup>1</sup>(''PGL''<sub>''n''</sub>)
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| in the [[projective linear group]], where ''n'' is the [[dimension of a variety|dimension]] of ''V''. There is a [[short exact sequence]]
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| :1 → ''GL''<sub>1</sub> → ''GL''<sub>''n''</sub> → ''PGL''<sub>''n''</sub> → 1
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| of [[algebraic group]]s. This implies a [[connecting homomorphism]]
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| :''H''<sup>1</sup>(''PGL''<sub>''n''</sub>) → ''H''<sup>2</sup>(''GL''<sub>1</sub>)
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| at the level of cohomology. Here ''H''<sup>2</sup>(''GL''<sub>''1''</sub>) is identified with the [[Brauer group]] of ''K'', while the kernel is trivial because
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| :''H''<sup>1</sup>(''GL''<sub>''n''</sub>) = {1}
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| by an extension of [[Hilbert's Theorem 90]].<ref name=GS26>Gille & Szamuely (2006) p.26</ref><ref>{{citation | title=An Introduction to Galois Cohomology and its Applications | volume=377 | series=London Mathematical Society Lecture Note Series | first=Grégory | last=Berhuy | publisher=[[Cambridge University Press]] | year=2010 | isbn=0-521-73866-0 | zbl=1207.12003 | page=113 }}</ref> Therefore the Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of [[central simple algebra]]s.
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| Lichtenbaum showed that if ''X'' is a Severi–Brauer variety over ''K'' then there is an exact sequence
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| :<math>0 \rightarrow \mathrm{Pic}(X) \rightarrow \mathbb{Z} \stackrel{\delta}{\rightarrow} \mathrm{Br}(K) \rightarrow \mathrm{Br}(K)(X) \rightarrow 0 \ . </math> | |
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| Here the map δ sends 1 to the Brauer class corresponding to ''X''.<ref name="GS129"/>
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| 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''.<ref name=GS131>Gille & Szamuely (2006) p.131</ref>
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| ==References==
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| {{reflist}}
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| *{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981) | publisher=[[Springer-Verlag]] | location=Berlin, New York | others=Notes by A. Verschoren | series=Lecture Notes in Math. | isbn=978-3-540-11216-7 | doi=10.1007/BFb0092235 | mr=657430 | year=1982 | volume=917 | chapter=Brauer-Severi varieties | zbl=0536.14006 | pages=194–210}}
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| *{{Springer|id=b/b017620|title=Brauer–Severi variety}}
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| *{{citation | chapterurl=http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511607219&cid=CBO9780511607219A036 | pages=114–134 | title=Central Simple Algebras and Galois Cohomology | first1=Philippe | last1=Gille | first2=Tamás | last2=Szamuely | series=Cambridge Studies in Advanced Mathematics | volume=101 | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
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| *{{citation | last=Saltman | first=David J. | title=Lectures on division algebras | series=Regional Conference Series in Mathematics | volume=94 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1999 | isbn=0-8218-0979-2 | zbl=0934.16013 }}
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| *{{Citation | last1=Severi | first1=Francesco | title=Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietà algebrica | language=Italian | id=Reprinted in volume 3 of his collected works | year=1932 | journal=Memorie della Reale Accademia d'Italia | volume=3 | issue=5}}
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| ==External links==
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| *[http://www.mathcs.emory.edu/~brussel/Papers/galoisdescent.pdf Expository paper on Galois descent (PDF)]
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| {{DEFAULTSORT:Severi-Brauer Variety}}
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| [[Category:Algebraic varieties]]
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| [[Category:Diophantine geometry]]
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| [[Category:Homological algebra]]
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| [[Category:Algebraic groups]]
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| [[Category:Ring theory]]
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I'm Monroe and I live with my husband and our 2 children in Drachten, in the FRANCE south area. My hobbies are Metal detecting, Auto audiophilia and Bonsai.
Here is my blog post: Fifa 15 Coin Generator