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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?

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3 Answers 3

up vote 4 down vote accepted

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 $Z$.

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  1. I don't know. Do you have a reference for the affine case ?

  2. The variety obtained is called "integral closure" (or "normalization") of $V$ in $L$. ;-) A reference is EGA II 6.3, which seems pretty canonical. EGA II 6.3.4 tells you that for any scheme $X$, for any quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$, you can construct the integral closure of $X$ in $\mathcal{A}$, which is an affine scheme $X'$ over $X$.

  3. It seems that it should generalize trivially, maybe I am missing something ? If I understand your question correctly, you have a scheme $X$, a quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{A}$, the integral closure $X'\rightarrow X$ of $X$ in $\mathcal{A}$ and an étale map $f:Y\rightarrow X$. You define $\mathcal{B}$ to be the pull-back of $\mathcal{A}$ (i.e. $f^{-1}\mathcal{A}\otimes_{f^{-1}\mathcal{O}_X}\mathcal{O}_Y$), and you want to show that $Y':=X'\times_X Y$ is the integral closure of $Y$ in $\mathcal{B}$. You can assume that $X$ is affine. Then, for every affine open $U$ of $Y$, the proposition you quote tells you that $Y'_U\rightarrow U$ is the integral closure of $U$ in $\mathcal{B}_U$, which, if I understand EGA II 6.3.4, is just saying that $Y'$ is the integral closure of $Y$ in $\mathcal{B}$.

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1. The statement in the affine case is that finite base change of a Henselian pair yields a Henselian pair. So, base change of the 'smallest pro-etale neighbourhood' of $Z$ in $X$ yields the smallest pro-etale neighbourhood of $Z_L$ in $X_L$. –  Mikhail Bondarko Nov 16 '11 at 17:23
    
See the beginning of Gabber's springerlink.com/content/c3x37474162663hp for this statement –  Mikhail Bondarko Nov 16 '11 at 17:39
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The answers above completely answer your questions. Let me just point out the following useful reference (to me at least).

For question 2, you could look at Chapter 4.1.2. of Liu's Algebraic geometry and arithmetic curves. See Definition 4.1.24.

Using a standard argument with the trace form one can show that, for any integral normal noetherian scheme $X$ with function field $K(X)$, the normalization $X^\prime \to X$ of $X$ in a finite separable extension of $K(X)$ is a finite morphism (Proposition 4.1.25).

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