What is "restriction of scalars" for a torus? I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am completely baffled by this scalar restriction business of having a field extension $K/k$ , a torus over $K$ and "restricting" it to $k$.
I would be very gratefull for a reference or even better by some explanation . I found nothing in my standard books (Hartshorne, Qing Liu, Mumford etc) so I hope this question is appropriate for the site. Thank you.
 A: 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.)
A: There is another description for tori. The category of tori over a field F is equivalent to the category of finite-dimensional $G_F$-lattices. Now, there is an operation of induction for group representations that converts a  $G_K$-lattice into a  $G_k$-lattice; this is the lattice you need.
See http://en.wikipedia.org/wiki/Algebraic_torus
A: It's not that hard at all. Here is an example. Let $k = \mathbf R$ and $K = \mathbf C$.
Consider a 1-dimensional torus $G_m$ over $\mathbf C$. It basically the group $\mathbf C^*$ over $\mathbf C$. 
Now $G = Res_{\mathbf C/\mathbf R} G_m$ is the same group $\mathbf C^*$ considered as group over $\mathbf R$.
