In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
Definition Let L/k be a finite extension of fields, and X a variety defined over L. The functor Res L / k X {\displaystyle \operatorname {Res} _{L/k}X} from k-schemesop to sets is defined by
Res L / k X ( S ) = X ( S × k L ) {\displaystyle \operatorname {Res} _{L/k}X(S)=X(S\times _{k}L)}
(In particular, the k-rational points of Res L / k X {\displaystyle \operatorname {Res} _{L/k}X} 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 ) {\displaystyle \operatorname {Spec} (L)\to \operatorname {Spec} (k)} and is right adjoint to fiber product of schemes, 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.
Alternative definition Let h : S ′ → S {\displaystyle h:S'\to S} be a morphism of schemes. For a S ′ {\displaystyle S'} -scheme X {\displaystyle X} , if the contravariant functor
Res S ′ / S ( X ) : S c h / S o p → S e t , T ↦ Hom S ′ ( T × S S ′ , X ) {\displaystyle \operatorname {Res} _{S'/S}(X):\mathbf {Sch/S} ^{op}\to \mathbf {Set} ,\quad T\mapsto \operatorname {Hom} _{S'}(T\times _{S}S',X)}
is representable, then we call the corresponding S {\displaystyle S} -scheme, which we also denote with Res S ′ / S ( X ) {\displaystyle \operatorname {Res} _{S'/S}(X)} , the Weil restriction of X {\displaystyle X} with respect to h {\displaystyle h} . Where S c h / S o p {\displaystyle \mathbf {Sch/S} ^{op}} denotes the dual of the category of schemes over a fixed scheme S {\displaystyle S} .
Properties For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension. Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism T → S {\displaystyle T\to S} of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.
Examples and applications Simple examples are the following:
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