Weil restriction (${\mathrm{Res}}_{S^{\prime }/S}X$)

For a morphism of schemes $S^{\prime }\to S$ and an $S^{\prime }$-scheme $X$, the Weil restriction is the functor defined on $S$-schemes byWhen this functor is represented by a scheme, that representing object is also called the Weil restriction. Representability needs hypotheses; it is not part of a formal guarantee for every $X$ and every base map.^[ref]

For $\mathbb{C}/\mathbb{R}$, restricting the affine line gives ${\mathrm{Res}}_{\mathbb{C}/\mathbb{R}}{\mathbb{A}}_{\mathbb{C}}^1\cong {\mathbb{A}}_{\mathbb{R}}^2$. Indeed, a complex coordinate is expressed as $a+bi$, and for any real algebra $R$ the corresponding points are $R{\otimes }_{\mathbb{R}}\mathbb{C}\cong R^2$. The construction expresses geometry over an extension field using coordinates over the smaller field, while retaining its full point functor.