I have some questions such that the corresponding statements are well-known for affine varieties, and I wonder whether they hold for projective ones.


1. Let $Z\subset X$ be a closed subvariety of a (projective) variety over a field $K$. Let $L/K$ be a finite field extension, and let $Y/X\_L$ be an etale neighbourhood of $Z\_L$ in $X\_L$ i.e. $Y/X\_L$ is etale and $Y\times\_{X\_L}Z\_L=Z\_L$. Is it true that $Y$ descends to an etale neighbourhood $U$ of $Z$ in $X$ i.e. that there exists an (etale neighbourhood) $U$ such that the morphism $U\_L\to X\_L$ factorizes through $Y$? Would it help if I will demand that $L/K$ is separable or Galois?
2. A reference question. For a domain $R$ and an extension $L$ of the fraction field of $R$ one can consider the integral closure (or the normalization) of $R$ in $L$. Now, Theorem 5.1 here <http://mathsci.kaist.ac.kr/~jinhyun/note/normalization/normalization.pdf> yields that a similar fact holds for any (irreducible) variety $V$ (instead of the spectrum of $R$) and a finite extension of the function field of $V$. Is there a 'canonical' reference for this fact? How would you call the variety obtained?
3. The integral closure opeation for (commutative) rings commutes with etale base change by [EGAIV, Prop. 18.12.15]. Does this statement generalize to varieties?



Do etale neighhbourhoods of a subvariety descend along base field extensions; does normalization commute with etale base change?
For question 1, you can use Weil restriction. More generally, let $X'\to X$ be a finite locally free morphism of schemes (in the present case it will be $X\_L\to X$), and let $Y$ be an $X'$-scheme. Recall that the Weil restriction functor $U$ of $Y\to X'$ (relative to $X'\to X$) associates to an $X$-scheme $T$ the set 
 $$U(T):=\mathrm{Hom}\_{X'}(T\times\_X X',Y);$$ in other words, it is right adjoint to the base change functor. In particular we have a canonical $X'$-morphism $U\times\_X X'\to Y$. Now (see Bosch-Lütkebohmert-Raynaud Néron Models, 7.6):  

(a) If $Y\to X'$ is quasiprojective, then $U$ is (representable by) a quasiprojective $X$-scheme.  

(b) If $Y\to X'$ is smooth (resp. étale), so is $U\to X$.  

(c) This construction clearly commutes with every base change $Z\to X$. In particular, in your situation, $Y\times\_X Z\to Z':=X'\times\_X Z$ is an isomorphism, so $U\times\_X Z$ is the Weil restriction of $Z'$ relative to $Z'\to Z$, which is $Z$.


