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 RC/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 EL/K functor to which it is adjoint. For any K-algebra over a field A we have EL/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 RL/K is the right adjoint to EL/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 schemes 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 [1], 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.
Referenced By
List of algebraic geometry topics | List of mathematical topics (V-Z) | Several complex variables
|
