Revisions to What is "restriction of scalars" for a torus?

replaced http://www.math.uga.edu/~pete with http://alpha.math.uga.edu/~pete

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edited Jun 22, 2022 at 8:14

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As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf http://alpha.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$, as it should be: see e.g. p. 2 of

http://alpha.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

Corrected some wrong parentheses

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edit approved May 17, 2014 at 21:26

MichalisN

363
4
7

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\[x,y\]((x^2+y^2)^\{-1\})$$\mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\[x,y\]((x^2+y^2)^\{-1\})$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\left[x,y\right]\left[(x^2+y^2)^{-1}\right]$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

Texified, by request

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edited Jan 15, 2010 at 19:45

Charles Siegel

16k
8
89
134

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an n$n$-dimensional complex variety as a (2n)$(2n)$-dimensional real variety.

The setup is as follows: let L/K$L/K$ be a finite degree field extension and let X$X$ be a scheme over L$L$. Then the Weil restriction W_{L/K} X$W_{L/K} X$ is the K$K$-scheme representing the following functor on on the category of K-algebras:

A |-> X(A \otimes_K L)$A\mapsto X(A \otimes_K L)$.

In particular, one has W_{L/K} X(K) = X(L)$W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever X$X$ is reduced of finite type.

Now for a more concrete description. Suppose X = Spec L[y_1,...,y_n]/J$X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let d = [L:K]$d = [L:K]$ and a_1,...,a_d$a_1,...,a_d$ be a K$K$-basis of L$L$. Then we make the following "substitution":

y_i = a_1 x_{i1} + ... + a_d x_{id},$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each y_i$y_i$ by a linear expression in d new variables x_{ij}$x_{ij}$. Moreover, suppose J = < g_1,...,g_m >;$J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into g_k(y_1,...,y_n)$g_k(y_1,...,y_n)$ getting a polynomial in the x$x$-variables, however still with L$L$-coefficients. But now using our fixed basis of L/K$L/K$, we can regard a single polynomial with L with $L$-coefficients as a vector of d$d$ polynomials with K$K$ coefficients. Thus we end up with md with $md$ generating polynomials in the x$x$-variables, say generating an ideal I$I$ in K[x_{ij}] $K[x_{ij}]$, and we put Res_{L/K} X = Spec K[x_{ij}]/I$\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case X = G_m$X = G_m$ (multiplicative group) over L = C$L = \mathbb{C}$ (complex numbers) and K = R$K = \mathbb{R}$. Then X$X$ is the spectrum of

C[y_1,.y_2]/(y_1 y_2 - 1);$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put y_i = x_{i1} + \sqrt{-1} x_{i2}$y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is R[x,y]((x^2+y^2)^{-1})$\mathbb{R}\[x,y\]((x^2+y^2)^\{-1\})$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety X/L$X/L$, the dimension of the Weil restriction from L$L$ down to K$K$ is [L:K]$[L:K]$ times the dimension of X/L$X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map X -> Spec L$X \to \mathrm{Spec} L$ with the map Spec L - > Spec K$\mathrm{Spec} L \to Spec K$ to give a map X -> Spec K$X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an n-dimensional complex variety as a (2n)-dimensional real variety.

The setup is as follows: let L/K be a finite degree field extension and let X be a scheme over L. Then the Weil restriction W_{L/K} X is the K-scheme representing the following functor on the category of K-algebras:

A |-> X(A \otimes_K L).

In particular, one has W_{L/K} X(K) = X(L).

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever X is reduced of finite type.

Now for a more concrete description. Suppose X = Spec L[y_1,...,y_n]/J is an affine scheme. Let d = [L:K] and a_1,...,a_d be a K-basis of L. Then we make the following "substitution":

y_i = a_1 x_{i1} + ... + a_d x_{id},

thus replacing each y_i by a linear expression in d new variables x_{ij}. Moreover, suppose J = < g_1,...,g_m >; then we substitute each of the above equations into g_k(y_1,...,y_n) getting a polynomial in the x-variables, however still with L-coefficients. But now using our fixed basis of L/K, we can regard a single polynomial with L-coefficients as a vector of d polynomials with K coefficients. Thus we end up with md generating polynomials in the x-variables, say generating an ideal I in K[x_{ij}], and we put Res_{L/K} X = Spec K[x_{ij}]/I.

A great example to look at is the case X = G_m (multiplicative group) over L = C (complex numbers) and K = R. Then X is the spectrum of

C[y_1,.y_2]/(y_1 y_2 - 1);

put y_i = x_{i1} + \sqrt{-1} x_{i2} and do the algebra. You can really see that the corresponding real affine variety is R[x,y]((x^2+y^2)^{-1}), as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety X/L, the dimension of the Weil restriction from L down to K is [L:K] times the dimension of X/L. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map X -> Spec L with the map Spec L - > Spec K to give a map X -> Spec K. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)

As said, the sought after concept is also known as Weil restriction. In a word, it is the algebraic analogue of the process of viewing an $n$-dimensional complex variety as a $(2n)$-dimensional real variety.

The setup is as follows: let $L/K$ be a finite degree field extension and let $X$ be a scheme over $L$. Then the Weil restriction $W_{L/K} X$ is the $K$-scheme representing the following functor on the category of K-algebras:

$A\mapsto X(A \otimes_K L)$.

In particular, one has $W_{L/K} X(K) = X(L)$.

By abstract nonsense (Yoneda...), if such a scheme exists it is uniquely determined by the above functor. For existence, some hypotheses are necessary, but I believe that it exists whenever $X$ is reduced of finite type.

Now for a more concrete description. Suppose $X = \mathrm{Spec} L[y_1,...,y_n]/J$ is an affine scheme. Let $d = [L:K]$ and $a_1,...,a_d$ be a $K$-basis of $L$. Then we make the following "substitution":

$$y_i = a_1 x_{i1} + ... + a_d x_{id},$$

thus replacing each $y_i$ by a linear expression in d new variables $x_{ij}$. Moreover, suppose $J = \langle g_1,...,g_m \rangle$; then we substitute each of the above equations into $g_k(y_1,...,y_n)$ getting a polynomial in the $x$-variables, however still with $L$-coefficients. But now using our fixed basis of $L/K$, we can regard a single polynomial with $L$-coefficients as a vector of $d$ polynomials with $K$ coefficients. Thus we end up with $md$ generating polynomials in the $x$-variables, say generating an ideal $I$ in $K[x_{ij}]$, and we put $\mathrm Res_{L/K} X = \mathrm{Spec} K[x_{ij}]/I$.

A great example to look at is the case $X = G_m$ (multiplicative group) over $L = \mathbb{C}$ (complex numbers) and $K = \mathbb{R}$. Then $X$ is the spectrum of

$$\mathbb{C}[y_1,y_2]/(y_1 y_2 - 1);$$

put $y_i = x_{i1} + \sqrt{-1} x_{i2}$ and do the algebra. You can really see that the corresponding real affine variety is $\mathbb{R}\[x,y\]((x^2+y^2)^\{-1\})$, as it should be: see e.g. p. 2 of

http://www.math.uga.edu/~pete/SC5-AlgebraicGroups.pdf

for the calculations.

Note the important general property that for a variety $X/L$, the dimension of the Weil restriction from $L$ down to $K$ is $[L:K]$ times the dimension of $X/L$. This is good to keep in mind so as not to confuse it with another possible interpretation of "restriction of scalars", namely composition of the map $X \to \mathrm{Spec} L$ with the map $\mathrm{Spec} L \to Spec K$ to give a map $X \to \mathrm{Spec} K$. This is a much weirder functor, which preserves the dimension but screws up things like geometric integrality. (When I first heard about "restriction of scalars", I guessed it was this latter thing and got very confused.)