I am currently trying to get familiar with the Weil Restriction functor.

For a finite field extension $L|K$ it associates a variety over $K$ to every variety $X$ over $L$ as the representing object of the functor defined via $Res_{L|K}X(A)=X(A\otimes_K L)$ for every $K$-Algebra $A$. There is also an explicit description in terms of defining equations for the variety.

I have the following questions:

$1$) Is there a characterisation of the (essential) image of this functor? i.e. can we say which varieties over $K$ "come from" $L$?

$2$) Is Restriction of scalars "injective" i.e. if a $K$-variety admits the structure of an $L$-variety is this structure unique? (up to isomorphism)

$3$) In the case of $L=\mathbb{C}$ and $K=\mathbb{R}$ there are also analytification functors and we can "restrict scalars" on the analytic side by regarding a complex manifold as a real analytic one. It seems to me, by the explicit construction of the Weil-Restriction, that analytification and restriction of scalars commute. Is this correct?

$3.5$) Can you recommend a reference to learn about this stuff which does not focus only on algebraic groups?



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