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An [[Algebraic variety|algebraic variety]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176201.png" /> that, if considered over the algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176202.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176203.png" />, becomes isomorphic to a [[Projective space|projective space]].
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The arithmetic properties of such varieties were studied in 1932 by F. Severi; F. Châtelet subsequently discovered a connection between Brauer–Severi varieties and central simple algebras (cf. [[Central simple algebra|Central simple algebra]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176204.png" /> and the [[Brauer group|Brauer group]].
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The simplest non-trivial example of a one-dimensional Brauer–Severi variety is the projective conic section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176205.png" />:
+
An [[Algebraic variety|algebraic variety]] over a field  $  k $
 +
that, if considered over the algebraic closure  $  \overline{k}\; $
 +
of $  k $,
 +
becomes isomorphic to a [[Projective space|projective space]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176206.png" /></td> </tr></table>
+
The arithmetic properties of such varieties were studied in 1932 by F. Severi; F. Châtelet subsequently discovered a connection between Brauer–Severi varieties and central simple algebras (cf. [[Central simple algebra|Central simple algebra]]) over  $  k $
 +
and the [[Brauer group|Brauer group]].
  
on the real projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176207.png" />. Over the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176208.png" /> this variety is isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b0176209.png" />. The set of all one-dimensional Brauer–Severi varieties, considered up to isomorphism, is in a one-to-one correspondence with the set of projective non-degenerate conic sections (considered up to projective equivalence over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762010.png" />), which is in turn in a one-to-one correspondence with the set of non-isomorphic generalized [[Quaternion|quaternion]] algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762011.png" />. In the above example the conical section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762012.png" /> corresponds to the algebra of ordinary quaternions.
+
The simplest non-trivial example of a one-dimensional Brauer–Severi variety is the projective conic section $  Q $:
  
In the more-dimensional case, the set of classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762013.png" />-dimensional Brauer–Severi varieties (i.e. Brauer–Severi varieties distinguished up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762014.png" />-isomorphism) may be identified with the [[Galois cohomology|Galois cohomology]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762015.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762016.png" /> is the projective group of automorphisms of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762017.png" /> [[#References|[3]]], [[#References|[4]]]. This cohomology group describes the classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762018.png" />-isomorphic central simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762019.png" />-algebras of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762020.png" /> (i.e. forms of the matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762021.png" />). The connection between Brauer–Severi varieties and central simple algebras is more explicitly described as follows. To a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762022.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762023.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762024.png" /> one associates the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762025.png" /> of its left ideals of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762026.png" />, which is defined as a closed subvariety of the [[Grassmann manifold|Grassmann manifold]] of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762027.png" />-linear subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762029.png" />. In certain cases the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762030.png" /> may be defined by norm equations — e.g. in the case of quaternion algebras. The connection between Brauer–Severi varieties and algebras is taken advantage of in the study of both [[#References|[1]]], [[#References|[4]]].
+
$$
 +
x _ {0}  ^ {2} + x _ {1}  ^ {2} +
 +
x _ {2}  ^ {2}  = 0
 +
$$
  
The most significant properties of Brauer–Severi varieties are the following. A Brauer–Severi variety is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762031.png" />-isomorphic to a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762032.png" /> if and only if it has a point in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762033.png" />. All Brauer–Severi varieties have a point in some finite separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762035.png" /> [[#References|[1]]].
+
on the real projective plane  $  \mathbf P _ {\mathbf R }  ^ {2} $.
 +
Over the field of complex numbers  $  \mathbf C $
 +
this variety is isomorphic to the projective line  $  \mathbf P _ {\mathbf C }  ^ {1} $.
 +
The set of all one-dimensional Brauer–Severi varieties, considered up to isomorphism, is in a one-to-one correspondence with the set of projective non-degenerate conic sections (considered up to projective equivalence over  $  k $),
 +
which is in turn in a one-to-one correspondence with the set of non-isomorphic generalized [[Quaternion|quaternion]] algebras over  $  k $.
 +
In the above example the conical section  $  Q $
 +
corresponds to the algebra of ordinary quaternions.
 +
 
 +
In the more-dimensional case, the set of classes of  $  n $-
 +
dimensional Brauer–Severi varieties (i.e. Brauer–Severi varieties distinguished up to  $  k $-
 +
isomorphism) may be identified with the [[Galois cohomology|Galois cohomology]] group  $  H  ^ {1} (k,  \mathop{\rm PGL} (n + 1, k)) $
 +
where  $  \mathop{\rm PGL} (n + 1, k) $
 +
is the projective group of automorphisms of the projective space  $  \textrm{ P } _ {k}  ^ {n} $[[#References|[3]]], [[#References|[4]]]. This cohomology group describes the classes of  $  k $-
 +
isomorphic central simple  $  k $-
 +
algebras of rank  $  (n + 1)  ^ {2} $(
 +
i.e. forms of the matrix algebra  $  M _ {n+1} (k) $).
 +
The connection between Brauer–Severi varieties and central simple algebras is more explicitly described as follows. To a  $  k $-
 +
algebra  $  A $
 +
of rank  $  r  ^ {2} $
 +
one associates the variety  $  X $
 +
of its left ideals of rank  $  r $,
 +
which is defined as a closed subvariety of the [[Grassmann manifold|Grassmann manifold]] of all  $  k $-
 +
linear subspaces of dimension  $  r $
 +
in  $  A $.
 +
In certain cases the variety  $  X $
 +
may be defined by norm equations — e.g. in the case of quaternion algebras. The connection between Brauer–Severi varieties and algebras is taken advantage of in the study of both [[#References|[1]]], [[#References|[4]]].
 +
 
 +
The most significant properties of Brauer–Severi varieties are the following. A Brauer–Severi variety is $  k $-
 +
isomorphic to a projective space $  \mathbf P _ {k}  ^ {n} $
 +
if and only if it has a point in the field $  k $.  
 +
All Brauer–Severi varieties have a point in some finite separable extension $  K $
 +
of $  k $[[#References|[1]]].
  
 
The [[Hasse principle|Hasse principle]] applies to Brauer–Severi varieties defined over an algebraic number field.
 
The [[Hasse principle|Hasse principle]] applies to Brauer–Severi varieties defined over an algebraic number field.
  
The field of rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762036.png" /> on a Brauer–Severi variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762037.png" /> is the splitting field of the corresponding algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762038.png" />; moreover, an arbitrary extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762040.png" /> is the splitting field for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762041.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762042.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762043.png" />-point [[#References|[4]]].
+
The field of rational functions $  k(X) $
 +
on a Brauer–Severi variety $  X $
 +
is the splitting field of the corresponding algebra $  A $;  
 +
moreover, an arbitrary extension $  K $
 +
of $  k $
 +
is the splitting field for $  A $
 +
if and only if $  X $
 +
has a $  K $-
 +
point [[#References|[4]]].
  
In the context of the generalization of the concepts of a central simple algebra and the Brauer group to include schemes, the Brauer–Severi varieties were generalized to the concept of Brauer–Severi schemes [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762044.png" /> be a morphism of schemes. A scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762045.png" /> is called a Brauer–Severi scheme if it is locally isomorphic to a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762046.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762047.png" /> in the [[Etale topology|étale topology]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762048.png" />. A scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762049.png" /> over a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762050.png" /> is a Brauer–Severi scheme if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762051.png" /> is a finitely-presented proper flat morphism and if all of its geometrical fibres are isomorphic to projective spaces [[#References|[2]]].
+
In the context of the generalization of the concepts of a central simple algebra and the Brauer group to include schemes, the Brauer–Severi varieties were generalized to the concept of Brauer–Severi schemes [[#References|[2]]]. Let $  f: P \rightarrow X $
 +
be a morphism of schemes. A scheme $  P $
 +
is called a Brauer–Severi scheme if it is locally isomorphic to a projective space $  {\mathbf P } _ {X}  ^ {n} $
 +
over $  X $
 +
in the [[Etale topology|étale topology]] of $  X $.  
 +
A scheme $  P $
 +
over a scheme $  X $
 +
is a Brauer–Severi scheme if and only if $  f: P \rightarrow X $
 +
is a finitely-presented proper flat morphism and if all of its geometrical fibres are isomorphic to projective spaces [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Châtelet,  "Variations sur un thème de H. Poincaré"  ''Ann. Sci. École Norm. Sup. (3)'' , '''61'''  (1944)  pp. 249–300</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Le groupe de Brauer"  A. Grothendieck (ed.)  J. Giraud (ed.)  et al. (ed.) , ''Dix exposés sur la cohomologie des schémas'' , North-Holland &amp; Masson  (1968)  pp. 1–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Cohomologie Galoisienne" , Springer  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Roguette,  "On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras"  ''Math. Ann.'' , '''150'''  (1963)  pp. 411–439</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Châtelet,  "Variations sur un thème de H. Poincaré"  ''Ann. Sci. École Norm. Sup. (3)'' , '''61'''  (1944)  pp. 249–300</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Le groupe de Brauer"  A. Grothendieck (ed.)  J. Giraud (ed.)  et al. (ed.) , ''Dix exposés sur la cohomologie des schémas'' , North-Holland &amp; Masson  (1968)  pp. 1–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Cohomologie Galoisienne" , Springer  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Roguette,  "On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras"  ''Math. Ann.'' , '''150'''  (1963)  pp. 411–439</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Thus a Brauer–Severi variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762052.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762053.png" />-form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017620/b01762054.png" />.
+
Thus a Brauer–Severi variety of dimension $  n $
 +
is a $  \overline{k}\; /k $-
 +
form of $  \mathbf P _ {k}  ^ {n} $.

Revision as of 06:29, 30 May 2020


An algebraic variety over a field $ k $ that, if considered over the algebraic closure $ \overline{k}\; $ of $ k $, becomes isomorphic to a projective space.

The arithmetic properties of such varieties were studied in 1932 by F. Severi; F. Châtelet subsequently discovered a connection between Brauer–Severi varieties and central simple algebras (cf. Central simple algebra) over $ k $ and the Brauer group.

The simplest non-trivial example of a one-dimensional Brauer–Severi variety is the projective conic section $ Q $:

$$ x _ {0} ^ {2} + x _ {1} ^ {2} + x _ {2} ^ {2} = 0 $$

on the real projective plane $ \mathbf P _ {\mathbf R } ^ {2} $. Over the field of complex numbers $ \mathbf C $ this variety is isomorphic to the projective line $ \mathbf P _ {\mathbf C } ^ {1} $. The set of all one-dimensional Brauer–Severi varieties, considered up to isomorphism, is in a one-to-one correspondence with the set of projective non-degenerate conic sections (considered up to projective equivalence over $ k $), which is in turn in a one-to-one correspondence with the set of non-isomorphic generalized quaternion algebras over $ k $. In the above example the conical section $ Q $ corresponds to the algebra of ordinary quaternions.

In the more-dimensional case, the set of classes of $ n $- dimensional Brauer–Severi varieties (i.e. Brauer–Severi varieties distinguished up to $ k $- isomorphism) may be identified with the Galois cohomology group $ H ^ {1} (k, \mathop{\rm PGL} (n + 1, k)) $ where $ \mathop{\rm PGL} (n + 1, k) $ is the projective group of automorphisms of the projective space $ \textrm{ P } _ {k} ^ {n} $[3], [4]. This cohomology group describes the classes of $ k $- isomorphic central simple $ k $- algebras of rank $ (n + 1) ^ {2} $( i.e. forms of the matrix algebra $ M _ {n+1} (k) $). The connection between Brauer–Severi varieties and central simple algebras is more explicitly described as follows. To a $ k $- algebra $ A $ of rank $ r ^ {2} $ one associates the variety $ X $ of its left ideals of rank $ r $, which is defined as a closed subvariety of the Grassmann manifold of all $ k $- linear subspaces of dimension $ r $ in $ A $. In certain cases the variety $ X $ may be defined by norm equations — e.g. in the case of quaternion algebras. The connection between Brauer–Severi varieties and algebras is taken advantage of in the study of both [1], [4].

The most significant properties of Brauer–Severi varieties are the following. A Brauer–Severi variety is $ k $- isomorphic to a projective space $ \mathbf P _ {k} ^ {n} $ if and only if it has a point in the field $ k $. All Brauer–Severi varieties have a point in some finite separable extension $ K $ of $ k $[1].

The Hasse principle applies to Brauer–Severi varieties defined over an algebraic number field.

The field of rational functions $ k(X) $ on a Brauer–Severi variety $ X $ is the splitting field of the corresponding algebra $ A $; moreover, an arbitrary extension $ K $ of $ k $ is the splitting field for $ A $ if and only if $ X $ has a $ K $- point [4].

In the context of the generalization of the concepts of a central simple algebra and the Brauer group to include schemes, the Brauer–Severi varieties were generalized to the concept of Brauer–Severi schemes [2]. Let $ f: P \rightarrow X $ be a morphism of schemes. A scheme $ P $ is called a Brauer–Severi scheme if it is locally isomorphic to a projective space $ {\mathbf P } _ {X} ^ {n} $ over $ X $ in the étale topology of $ X $. A scheme $ P $ over a scheme $ X $ is a Brauer–Severi scheme if and only if $ f: P \rightarrow X $ is a finitely-presented proper flat morphism and if all of its geometrical fibres are isomorphic to projective spaces [2].

References

[1] F. Châtelet, "Variations sur un thème de H. Poincaré" Ann. Sci. École Norm. Sup. (3) , 61 (1944) pp. 249–300
[2] A. Grothendieck, "Le groupe de Brauer" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 1–21
[3] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964)
[4] P. Roguette, "On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras" Math. Ann. , 150 (1963) pp. 411–439

Comments

Thus a Brauer–Severi variety of dimension $ n $ is a $ \overline{k}\; /k $- form of $ \mathbf P _ {k} ^ {n} $.

How to Cite This Entry:
Brauer-Severi variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer-Severi_variety&oldid=46157
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article