= Weil restriction
{c}
{title2=$\operatorname{Res}_{S'/S}X$}
For a <morphism of schemes> $S'\to S$ and an $S'$-<scheme> $X$, the <Weil restriction> is the <functor> defined on $S$-<schemes> by
$$
\operatorname{Res}_{S'/S}(X)(T)=\operatorname{Hom}_{S'}(T\times_S S',X).
$$
When 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.<The Stacks project>[tag 05Y8, Restriction of scalars, defining functor and representability results]{ref}
For $\mathbb C/\mathbb R$, restricting the affine line gives $\operatorname{Res}_{\mathbb C/\mathbb R}\mathbb A^1_{\mathbb C}\cong\mathbb A^2_{\mathbb R}$. 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.
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