Editing Weil restriction

{{Short description|Restriction of scalars}}
In [[mathematics]], '''restriction of scalars''' (also known as "[[André_Weil|Weil]] restriction") is a [[functor]] which, for any finite [[field extension|extension]] of [[field (mathematics)|field]]s ''L/k'' and any [[algebraic variety]] ''X'' over ''L'', produces another variety Res<sub>''L''/''k''</sub>''X'', 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 <math>\operatorname{Res}_{L/k} X </math> from ''k''-[[scheme (mathematics)|schemes]]<sup>op</sup> to sets is defined by

:<math>\operatorname{Res}_{L/k}X(S) = X(S \times_k L)</math>

(In particular, the ''k''-rational points of <math>\operatorname{Res}_{L/k}X</math> are the ''L''-rational points of ''X''.) The variety that [[representable functor|represents]] this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.

From the standpoint of [[sheaf (mathematics)|sheaves]] of sets, restriction of scalars is just a pushforward along the morphism <math>\operatorname{Spec}(L) \to \operatorname{Spec}(k)</math> 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 [[topos|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 <math>h:S'\to S</math> be a [[morphism of schemes]]. For a <math>S'</math>-scheme <math>X</math>, if the contravariant functor
:<math>\operatorname{Res}_{S'/S}(X):\mathbf{Sch/S}^{op}\to \mathbf{Set},\quad T\mapsto \operatorname{Hom}_{S'}(T\times_S S',X)</math>
is [[Representable functor|representable]], then we call the corresponding <math>S</math>-scheme, which we also denote with <math>\operatorname{Res}_{S'/S}(X)</math>, the Weil restriction of <math>X</math> with respect to <math>h</math>.<ref>{{cite book|first1=Siegfried|last1=Bosch|first2=Werner|last2=Lütkebohmert|first3=Michel|last3=Raynaud |publisher=Springer-Verlag |title=Néron models |location=Berlin |date=1990 |page=191}}</ref> 

Where <math>\mathbf{Sch/S}^{op}</math> denotes the [[dual category|dual]] of the category of schemes over a fixed scheme <math>S</math>.

== 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 <math>T \to S</math> of [[algebraic space]]s yields a restriction of scalars functor that takes [[algebraic stack]]s to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.

== Examples and applications ==
Simple examples are the following: 
# Let ''L'' be a finite extension of ''k'' of degree ''s''. Then <math>\operatorname{Res}_{L/k}(\operatorname{Spec} (L)) = \operatorname{Spec}(k)</math> and <math>\operatorname{Res}_{L/k}\mathbb{A}^1</math> is an ''s''-dimensional affine space <math> \mathbb{A}^s</math> over Spec ''k''.
# If ''X'' is an affine ''L''-variety, defined by <math display=block>X = \operatorname{Spec} L[x_1, \dots, x_n]/(f_1,\dotsc,f_m); </math>we can write <math>\operatorname{Res}_{L/k}X</math> as Spec <math>k[y_{i,j}]/(g_{l,r})</math>, where <math>y_{i,j}</math> (<math>1 \leq i \leq n, 1 \leq j \leq s</math>) are new variables, and <math>g_{l,r}</math> (<math>1 \leq l \leq m, 1 \leq r \leq s</math>) are polynomials in <math>y_{i,j}</math> given by taking a ''k''-basis <math>e_1, \dotsc, e_s</math> of ''L'' and setting <math>x_i = y_{i,1}e_1 + \dotsb + y_{i,s}e_s</math> and <math>f_t = g_{t,1}e_1 + \dotsb + g_{t,s}e_s</math>.

If a scheme is a [[group scheme]] then any Weil restriction of it will be as well. This is frequently used in [[number theory]], for instance: 
# The torus <math display=block>\mathbb{S} := \operatorname{Res}_{\Complex/\R} \mathbb{G}_m</math> where <math>\mathbb{G}_m</math> denotes the multiplicative group, plays a significant role in [[Hodge theory]], since the [[Tannakian category]] of real [[Hodge structure]]s is equivalent to the [[category of representations]] of <math>\mathbb{S}.</math> The real points have a [[Lie group]] structure isomorphic to <math>\Complex^\times</math>. See [[Mumford–Tate group]].
# The Weil restriction <math>\operatorname{Res}_{L/k} \mathbb{G}</math> of a (commutative) group variety <math>\mathbb{G}</math> is again a (commutative) group variety of dimension <math>[L:k]\dim \mathbb{G},</math> if ''L'' is separable over ''k''.
# Restriction of scalars on [[abelian variety|abelian varieties]] (e.g. [[elliptic curve]]s) yields abelian varieties, if ''L'' is separable over ''k''. James Milne used this to reduce the [[Birch and Swinnerton-Dyer conjecture]] for abelian varieties over all [[algebraic number field|number fields]] to the same conjecture over the rationals.
# In [[elliptic curve cryptography]], the [[Weil descent]] attack uses the Weil restriction to transform a [[discrete logarithm problem]] on an [[elliptic curve]] over a finite extension field L/K, into a discrete log problem on the [[Jacobian variety]] of a [[hyperelliptic curve]] over the base field K, that is potentially easier to solve because of K's smaller size.

== Weil restrictions vs. Greenberg transforms ==

Restriction of scalars is similar to the Greenberg transform, but does not generalize it, since the ring of [[Witt vector]]s on a commutative algebra ''A'' is not in general an ''A''-algebra.

== References ==
{{Reflist}}
The original reference is Section 1.3 of Weil's 1959-1960 Lectures, published as:

* Andre Weil. "Adeles and Algebraic Groups", Progress in Math. '''23''', Birkhäuser 1982. Notes of Lectures given 1959-1960.

Other references:

* Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud. "Néron models", Springer-Verlag, Berlin 1990.
* James S. Milne. "On the arithmetic of abelian varieties", Invent. Math. '''17''' (1972) 177-190.
* Martin Olsson. "Hom stacks and restriction of scalars", Duke Math J., '''134''' (2006), 139–164. http://math.berkeley.edu/~molsson/homstackfinal.pdf
* Bjorn Poonen. "Rational points on varieties", http://math.mit.edu/~poonen/papers/Qpoints.pdf
* [[Nigel Smart (cryptographer)|Nigel Smart]], Weil descent page with bibliography, https://homes.esat.kuleuven.be/~nsmart/weil_descent.html

[[Category:Algebraic varieties]]
[[Category:Scheme theory]]