View source for Brauer-Severi variety

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]].

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]].

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" />:

<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>

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.

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]]].

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]]].

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]]].

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]]].

====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>



====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" />.