Weil restriction

In mathematics, specifically the theory of algebraic groups, Weil restriction is a functor allowing one to pass from an algebraic group G over a field L to another one, RG , over a subfield (mathematics) K. The idea is that the group of points G ( L ) of G over L should be deemed RG ( K ).

For example taking L = C to be the complex number field, and K = R the real number field, we can apply Weil restriction to the multiplicative group

: GL 1

to get

: R C/RGL1,

which is a two-dimensional algebraic group. It consists of 2×2 matrices of the shape that is given by the action of a + bi on the basis {1, i } of C over R:

:egin{bmatrix} a & b \ -b & a end{bmatrix}. This group is an algebraic torus, and is applied in Hodge theory, where its linear representations are Hodge structures .

Note that the construction is of an . There is an extension of scalars

: E L / K

functor to which it is adjoint. For any K -algebra over a field A we have

: E L / K ( A )

the tensor product of A with L over K (as K -vector spaces), which is made into an L -algebra using the existing ring product in A and in L . Then it is almost true to say that R L / K is the right adjoint to E L / K .

To be completely accurate, we should do this: an algebraic group H over K is such that for a commutative K -algebra B , H ( B ) is

:Hom (Spec( B ), H )

in a suitable category (of scheme (mathematics)s over Spec( K )). Another way of putting it is that Spec makes the category of commutative K -algebras into its dual (category theory). Therefore the actual adjunction relation is of the type

:Hom( E Spec( B ), G ) = Hom(Spec( B ), RG )

where on the left side we are in the opposite of the category of commutative L -algebras, on the right side in the opposite of the category of commutative K -algebras, and E becomes the fiber product over Spec( K ) with Spec( L ). This is a complete definition in the case that G is an affine algebraic group.

The case where G is an 0).

The Weil restriction is essential for the classification of algebraic groups over fields that are not algebraically closed.