Weil Restriction - Definition

Definition

Let L/k be a finite extension of fields, and X a variety defined over L. The functor from k-schemesop to sets is defined by

(In particular, the k-rational points of are the L-rational points of X.) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.

From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L Spec k and is right adjoint to fiber product, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.