Severi-Brauer variety
Encyclopedia
In mathematics
, a Severi–Brauer variety over a field
K is an algebraic variety
V which becomes isomorphic to projective space
over an algebraic closure
of K. Examples are conic section
s C: provided C is non-singular, it becomes isomorphic to the projective line
over any extension field L over which C has a point defined. The name is for Francesco Severi
and Richard Brauer
.
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
in the projective linear group
, where n is the dimension of V. There is a short exact sequence
of algebraic group
s. This implies a connecting homomorphism
at the level of cohomology. Here H2(GL1) is identified with the Brauer group
of K, while the kernel is trivial because
by an extension of Hilbert's Theorem 90
. Therefore the Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebra
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Severi–Brauer variety over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V which becomes isomorphic to projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
over an algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of K. Examples are conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s C: provided C is non-singular, it becomes isomorphic to the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
over any extension field L over which C has a point defined. The name is for Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
and Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
.
Such varieties are of interest not only in diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
, but also in Galois cohomology
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
. They represent (at least if K is a perfect field
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
) Galois cohomology classes in
- H1(PGLn)
in the projective linear group
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
, where n is the dimension of V. There is a short exact sequence
- 1 → GL1 → GLn → PGLn → 1
of algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s. This implies a connecting homomorphism
- H1(PGLn) → H2(GL1)
at the level of cohomology. Here H2(GL1) is identified with the Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
of K, while the kernel is trivial because
- H1(GLn) = {1}
by an extension of Hilbert's Theorem 90
Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...
. Therefore the Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebra
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...
s.