Weil restriction

In 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 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 and K=R, we can apply Weil restriction to GL₁ to get R_C/RGL₁, which is a two-dimensional algebraic group. It consists of 2x2 matrices of the shape that is given by the action of a+bi on the basis {1,i} of C over R:

(This group is applied in Hodge theory.) Note that the construction is of an algebraic variety, not just a set of points: a group object, not simply a group.

To say this more formally, we should identify RG as a right adjoint. 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 schemess over Spec(K)). Another way of putting it is that Spec makes the category of commutative K-algebras into its opposite. Therefore the actual adjunction relation is of the type

Hom (ESpec(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 abelian variety is also of importance, though. It is one non-trivial way to construct higher-dimensional abelian varieties from elliptic curves, for example. Weil restriction multiplies dimension by [L:K], as one can compute with the tangent space (in characteristic 0).

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