Weil restriction - Knowlpedia

Core idea

Let $S'\to S$ be a finite locally free morphism and let $X$ be an $S'$ -scheme. The Weil restriction $\operatorname{Res}_{S'/S}X$ , when it exists, is the $S$ -scheme representing the functor

T\longmapsto \operatorname{Hom}_{S'}(T\times_S S',X)

on $S$ -schemes. Equivalently, it is characterized by natural bijections

\operatorname{Hom}_S(T,\operatorname{Res}_{S'/S}X) \cong \operatorname{Hom}_{S'}(T_{S'},X).

Existence and structure

If $X$ is affine over $S'$ , the Weil restriction exists and is affine over $S$ . It also exists under standard broader hypotheses, including when $X$ is quasi-projective and $S'\to S$ is finite locally free. Constructions such as products and group laws transport through the representing property, so the Weil restriction of a group scheme is a group scheme.

Complex groups viewed over the real numbers

For a group scheme $G$ over $\mathbb C$ ,

\bigl(\operatorname{Res}_{\mathbb C/\mathbb R}G\bigr)(\mathbb R) \cong G(\mathbb C).

For example, the real points of $\operatorname{Res}_{\mathbb C/\mathbb R}SL_2$ form $SL(2,\mathbb C)_{\mathbb R}$ . This explains algebraically why its real dimension is twice the complex dimension.

After base change back to $\mathbb C$ , one obtains

\bigl(\operatorname{Res}_{\mathbb C/\mathbb R}G\bigr)_{\mathbb C} \cong G\times\overline G,

where $\overline G$ is the conjugate group scheme. This reflects $\mathbb C\otimes_{\mathbb R}\mathbb C\cong\mathbb C\times\mathbb C$ . Weil restriction is therefore a representable algebraic universal construction, not merely the smooth operation of forgetting a complex structure, though the two agree after taking real points and analytifying in this setting.

References

Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron Models, Springer, 1990, §7.6. Publisher record.
The Stacks Project Authors, More on Morphisms, Section 97.11, “Restriction of scalars.” Stacks Project.