Weil restriction

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 Res_L/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

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In this video, I want to familiarize you with the idea of a limit, which is a super important idea. It's really the idea that all of calculus is based upon. But despite being so super important, it's actually a really, really, really, really, really, really simple idea. So let me draw a function here, actually, let me define a function here, a kind of a simple function. So let's define f of x, let's say that f of x is going to be x minus 1 over x minus 1. And you might say, hey, Sal look, I have the same thing in the numerator and denominator. If I have something divided by itself, that would just be equal to 1. Can't I just simplify this to f of x equals 1? And I would say, well, you're almost true, the difference between f of x equals 1 and this thing right over here, is that this thing can never equal-- this thing is undefined when x is equal to 1. Because if you set, let me define it. Let me write it over here, if you have f of, sorry not f of 0, if you have f of 1, what happens. In the numerator, we get 1 minus 1, which is, let me just write it down, in the numerator, you get 0. And in the denominator, you get 1 minus 1, which is also 0. And so anything divided by 0, including 0 divided by 0, this is undefined. So you can make the simplification. You can say that this is you the same thing as f of x is equal to 1, but you would have to add the constraint that x cannot be equal to 1. Now this and this are equivalent, both of these are going to be equal to 1 for all other X's other than one, but at x equals 1, it becomes undefined. This is undefined and this one's undefined. So how would I graph this function. So let me graph it. So that, is my y is equal to f of x axis, y is equal to f of x axis, and then this over here is my x-axis. And then let's say this is the point x is equal to 1. This over here would be x is equal to negative 1. This is y is equal to 1, right up there I could do negative 1. but that matter much relative to this function right over here. And let me graph it. So it's essentially for any x other than 1 f of x is going to be equal to 1. So it's going to be, look like this. It's going to look like this, except at 1. At 1 f of x is undefined. So I'm going to put a little bit of a gap right over here, the circle to signify that this function is not defined. We don't know what this function equals at 1. We never defined it. This definition of the function doesn't tell us what to do with 1. It's literally undefined, literally undefined when x is equal to 1. So this is the function right over here. And so once again, if someone were to ask you what is f of 1, you go, and let's say that even though this was a function definition, you'd go, OK x is equal to 1, oh wait there's a gap in my function over here. It is undefined. So let me write it again. It's kind of redundant, but I'll rewrite it f of 1 is undefined. But what if I were to ask you, what is the function approaching as x equals 1. And now this is starting to touch on the idea of a limit. So as x gets closer and closer to 1. So as we get closer and closer x is to 1, what is the function approaching. Well, this entire time, the function, what's a getting closer and closer to. On the left hand side, no matter how close you get to 1, as long as you're not at 1, you're actually at f of x is equal to 1. Over here from the right hand side, you get the same thing. So you could say, and we'll get more and more familiar with this idea as we do more examples, that the limit as x and L-I-M, short for limit, as x approaches 1 of f of x is equal to, as we get closer, we can get unbelievably, we can get infinitely close to 1, as long as we're not at 1. And our function is going to be equal to 1, it's getting closer and closer and closer to 1. It's actually at 1 the entire time. So in this case, we could say the limit as x approaches 1 of f of x is 1. So once again, it has very fancy notation, but it's just saying, look what is a function approaching as x gets closer and closer to 1. Let me do another example where we're dealing with a curve, just so that you have the general idea. So let's say that I have the function f of x, let me just for the sake of variety, let me call it g of x. Let's say that we have g of x is equal to, I could define it this way, we could define it as x squared, when x does not equal, I don't know when x does not equal 2. And let's say that when x equals 2 it is equal to 1. So once again, a kind of an interesting function that, as you'll see, is not fully continuous, it has a discontinuity. Let me graph it. So this is my y equals f of x axis, this is my x-axis right over here. Let me draw x equals 2, x, let's say this is x equals 1, this is x equals 2, this is negative 1, this is negative 2. And then let me draw, so everywhere except x equals 2, it's equal to x squared. So let me draw it like this. So it's going to be a parabola, looks something like this, let me draw a better version of the parabola. So it'll look something like this. Not the most beautifully drawn parabola in the history of drawing parabolas, but I think it'll give you the idea. I think you know what a parabola looks like, hopefully. It should be symmetric, let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared. But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here. So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that. So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3. So when x is equal to 2, our function is equal to 1. So this is a bit of a bizarre function, but we can define it this way. You can define a function however you like to define it. And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2. You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that. You use g of x is equal to 1. So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared. So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here. And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x. Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos. As x gets closer and closer to 2, what is g of x approaching? So if you get to 1.9, and then 1.999, and then 1.999999, and then 1.9999999, what is g of x approaching. Or if you were to go from the positive direction. If you were to say 2.1, what's g of 2.1, what's g of 2.01, what's g of 2.001, what is that approaching as we get closer and closer to it. And you can see it visually just by drawing the graph. As g gets closer and closer to 2, and if we were to follow along the graph, we see that we are approaching 4. Even though that's not where the function is, the function drops down to 1. The limit of g of x as x approaches 2 is equal to 4. And you could even do this numerically using a calculator, and let me do that, because I think that will be interesting. So let me get the calculator out, let me get my trusty TI-85 out. So here is my calculator, and you could numerically say, OK, what's it going to approach as you approach x equals 2. So let's try 1.94, for x is equal to 1.9, you would use this top clause right over here. So you'd have 1.9 squared. And so you get 3.61, well what if you get even closer to 2, so 1.99, and once again, let me square that. Well now I'm at 3.96. What if I do 1.999, and I square that? I'm going to have 3.996. Notice I'm going closer, and closer, and closer to our point. And if I did, if I got really close, 1.9999999999 squared, what am I going to get to. It's not actually going to be exactly 4, this calculator just rounded things up, but going to get to a number really, really, really, really, really, really, really, really, really close to 4. And we can do something from the positive direction too. And it actually has to be the same number when we approach from the below what we're trying to approach, and above what we're trying to approach. So if we try to 2.1 squared, we get 4.4. If we do 2. let me go a couple of steps ahead, 2.01, so this is much closer to 2 now, squared. Now we are getting much closer to 4. So the closer we get to 2, the closer it seems like we're getting to 4. So once again, that's a numeric way of saying that the limit, as x approaches 2 from either direction of g of x, even though right at 2, the function is equal to 1, because it's discontinuous. The limit as we're approaching 2, we're getting closer, and closer, and closer to 4.

Definition

Let L/k be a finite extension of fields, and X a variety defined over L. The functor $\operatorname {Res} _{L/k}X$ from k-schemes^op to sets is defined by

\operatorname {Res} _{L/k}X(S)=X(S\times _{k}L)

(In particular, the k-rational points of $\operatorname {Res} _{L/k}X$ 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 $\operatorname {Spec} (L)\to \operatorname {Spec} (k)$ 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 $h:S'\to S$ be a morphism of schemes. For a $S'$ -scheme $X$ , if the contravariant functor

\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 $S$ -scheme, which we also denote with $\operatorname {Res} _{S'/S}(X)$ , the Weil restriction of $X$ with respect to $h$ .^[1]

Where $\mathbf {Sch/S} ^{op}$ denotes the dual of the category of schemes over a fixed scheme $S$ .

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 $T\to S$ 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:

Let L be a finite extension of k of degree s. Then $\operatorname {Res} _{L/k}(\operatorname {Spec} (L))=\operatorname {Spec} (k)$ and $\operatorname {Res} _{L/k}\mathbb {A} ^{1}$ is an s-dimensional affine space $\mathbb {A} ^{s}$ over Spec k.
If X is an affine L-variety, defined by $X=\operatorname {Spec} L[x_{1},\dots ,x_{n}]/(f_{1},\dotsc ,f_{m});$ we can write $\operatorname {Res} _{L/k}X$ as Spec $k[y_{i,j}]/(g_{l,r})$ , where $y_{i,j}$ ( $1\leq i\leq n,1\leq j\leq s$ ) are new variables, and $g_{l,r}$ ( $1\leq l\leq m,1\leq r\leq s$ ) are polynomials in $y_{i,j}$ given by taking a k-basis $e_{1},\dotsc ,e_{s}$ of L and setting $x_{i}=y_{i,1}e_{1}+\dotsb +y_{i,s}e_{s}$ and $f_{t}=g_{t,1}e_{1}+\dotsb +g_{t,s}e_{s}$ .

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 $\mathbb {S} :=\operatorname {Res} _{\mathbb {C} /\mathbb {R} }\mathbb {G} _{m}$ where $\mathbb {G} _{m}$ 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 $\mathbb {S} .$ The real points have a Lie group structure isomorphic to $\mathbb {C} ^{\times }$ . See Mumford–Tate group.
The Weil restriction $\operatorname {Res} _{L/k}\mathbb {G}$ of a (commutative) group variety $\mathbb {G}$ is again a (commutative) group variety of dimension $[L:k]\dim \mathbb {G} ,$ if L is separable over k.
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.