View source for Severi–Brauer variety

In [[Mathematics|mathematics]], a '''Severi–Brauer variety''' over a [[Field (mathematics)|field]] ''K'' is an [[Algebraic variety|algebraic variety]] ''V'' which becomes isomorphic to a [[Projective space|projective space]] over an algebraic closure of ''K''.  The varieties are associated to [[Central simple algebra|central simple algebra]]s in such a way that the algebra splits over ''K'' if and only if the variety has a rational point over ''K''.{{sfnp|ps=|Jacobson|1996|p=113}} {{harvs|txt|first=Francesco |last=Severi|year=1932}} studied these varieties,  and they are also named after [[Biography:Richard Brauer|Richard Brauer]] because of their close relation to the [[Brauer group]].
In dimension one, the Severi–Brauer varieties are [[Conic section|conics]].  The corresponding central simple algebras are the [[Quaternion algebra|quaternion algebra]]s.  The algebra {{nowrap|(''a'', ''b'')<sub>''K''</sub>}} corresponds to the conic {{nowrap|''C''(''a'', ''b'')}} with equation 
: <math>z^2 = ax^2 + by^2 \ </math>
and the algebra {{nowrap|(''a'', ''b'')<sub>''K''</sub>}} ''splits'', that is, {{nowrap|(''a'', ''b'')<sub>''K''</sub>}} is isomorphic to a [[Matrix ring|matrix algebra]] over ''K'', if and only if {{nowrap|''C''(''a'', ''b'')}} has a point defined over ''K'': this is in turn equivalent to {{nowrap|''C''(''a'', ''b'')}} being isomorphic to the [[Projective line|projective line]] over ''K''.{{sfnp|ps=|Jacobson|1996|p=113}}{{sfnp|ps=|Gille|Szamuely|2006|p=129}}

Such varieties are of interest not only in [[Diophantine geometry|diophantine geometry]], but also in [[Galois cohomology]]. They represent (at least if ''K'' is a [[Perfect field|perfect field]]) Galois cohomology classes in
''H''<sup>1</sup>(G(K<sup>s</sup>/K),PGL<sub>''n''</sub>),
where PGL<sub>''n''</sub>
is the [[Projective linear group|projective linear group]], and ''n'' is one more than the dimension of 
the variety ''V''. As usual in Galois cohomology, we often leave the <math>G(K^s/K)</math> implied. There is a short exact sequence
: 1 → GL<sub>1</sub> → GL<sub>''n''</sub> → PGL<sub>''n''</sub> → 1
of [[Algebraic group|algebraic group]]s. This implies a connecting homomorphism
: ''H''<sup>1</sup>(PGL<sub>''n''</sub>) → ''H''<sup>2</sup>(GL<sub>1</sub>)
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
''H''<sup>1</sup>(GL<sub>''n''</sub>) = {{mset|1}}
by an extension of [[Hilbert's Theorem 90]].{{sfnp|ps=|Gille|Szamuely|2006|p=26}}{{sfnp|ps=|Berhuy|2010|p=113}} Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of [[Central simple algebra|central simple algebra]]s.

Lichtenbaum showed that if ''X'' is a Severi–Brauer variety over ''K'' then there is an exact sequence
: <math>0 \rightarrow \mathrm{Pic}(X) \rightarrow \mathbb{Z} ~\stackrel{\delta}{\rightarrow}~ \mathrm{Br}(K) \rightarrow \mathrm{Br}(K)/(X) \rightarrow 0 \ . </math>

Here the map ''δ'' sends 1 to the Brauer class corresponding to ''X''.{{sfnp|ps=|Gille|Szamuely|2006|p=129}}

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 [[Physics:Linear system|linear system]] defines the ''d''-dimensional embedding of ''X'' over a splitting field ''L''.{{sfnp|ps=|Gille|Szamuely|2006|p=131}}

== See also ==
* [[Projective bundle]]

== Note ==
{{reflist}}

== References ==
{{refbegin}}
* {{citation | last1=Artin | first1=Michael |  title=Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981) | publisher=[[Physics:Springer-Verlag|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}}
* {{springer | id=b/b017620 | title=Brauer–Severi variety}}
* {{citation | chapter-url=http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511607219&cid=CBO9780511607219A036 | pages=114–134 | title=Central Simple Algebras and Galois Cohomology | chapter=Severi–Brauer varieties | first1=Philippe | last1=Gille | first2=Tamás | last2=Szamuely | series=Cambridge Studies in Advanced Mathematics | volume=101 | publisher={{wipe|Cambridge University Press}} | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 | mr=2266528}}
* {{citation | title=An Introduction to Galois Cohomology and its Applications | volume=377 | series=London Mathematical Society Lecture Note Series | first=Grégory | last=Berhuy | publisher={{wipe|Cambridge University Press}} | year=2010 | isbn=0-521-73866-0 | zbl=1207.12003 }}
* {{citation | last=Jacobson | first=Nathan |  title=Finite-dimensional division algebras over fields | zbl=0874.16002 | location=Berlin | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | isbn=3-540-57029-2 | year=1996 }}
* {{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  }}
* {{citation | last1=Severi | first1=Francesco | title=Un nuovo campo di ricerche nella geometria sopra una superficie e sopra una varietà algebrica | language=it | id=Reprinted in volume 3 of his collected works | year=1932 | journal=Memorie della Reale Accademia d'Italia | volume=3 | issue=5}}
{{refend}}

== Further reading ==
{{refbegin}}
* {{citation | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander |  last3=Rost | first3=Markus |  last4=Tignol | first4=Jean-Pierre | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=American Mathematical Society | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 | mr=1632779}}
{{refend}}

== External links ==
{{refbegin}}
* [http://www.mathcs.emory.edu/~brussel/Papers/galoisdescent.pdf Expository paper on Galois descent (PDF)]
{{refend}}

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