Restriction of scalars
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 from k-
schemesop to sets is defined by
(In particular, the k-rational points of 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 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 be a
morphism of schemes. For a -scheme , if the contravariant functor
is
representable, then we call the corresponding -scheme, which we also denote with , the Weil restriction of with respect to .
[1]
Where denotes the
dual of the category of schemes over a fixed scheme .
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 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:
- Let L be a finite extension of k of degree s. Then and is an s-dimensional affine space over Spec k.
- If X is an affine L-variety, defined by
we can write as Spec , where () are new variables, and () are polynomials in given by taking a k-basis of L and setting and .
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
where denotes the multiplicative group, plays a significant role in Hodge theory, since the
Tannakian category of real
Hodge structures is equivalent to the category of representations of The real points have a
Lie group structure isomorphic to . See
Mumford–Tate group.
- The Weil restriction of a (commutative) group variety is again a (commutative) group variety of dimension if L is separable over k. Aleksander Momot applied Weil restrictions of commutative group varieties with and in order to derive new results in transcendence theory which were based on the increase in algebraic dimension.[
citation needed]
- Restriction of scalars on
abelian varieties (e.g.
elliptic curves) 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
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 vectors on a commutative algebra A is not in general an A-algebra.
References
-
^ Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel (1990). Néron models. Berlin: Springer-Verlag. p. 191.
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, Weil descent page with bibliography,
https://homes.esat.kuleuven.be/~nsmart/weil_descent.html
- Aleksander Momot, "Density of rational points on commutative group varieties and small transcendence degree",
https://arxiv.org/abs/1011.3368