The usual way to define the Henselization $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, \mathfrak q)$ over all étale neighborhoods of $A$ (i.e. pairs $(B,\mathfrak q)$ where $B$ is an étale $A$-algebra, prime ideal $q$ lies over $\mathfrak{m}$ such that the induced map $k:=A/\mathfrak{m} \to \kappa(\mathfrak q)= B_\mathfrak q/\mathfrak q$ is an isomorphism).
James Milne gave in his book Étale Cohomology at page 37 in Example 4.10 an alternative construction for Henselisation for the case of $A$ normal as follows:
Let $K$ be the field of fractions of $A$, and $K_\text{sep}$ be a separable closure of $K$. The Galois group $G$ of $K_\text{sep}$ over $K$ acts on the integral closure $B$ of $A$ in $K_\text{sep}$. Let $\mathfrak n$ be a maximal ideal of $B$ lying over $\mathfrak m$, and let $D \subset G$ be the decomposition group of it, that is, $D = \{\sigma \in G \ \vert \ \sigma(n)=n \}$. Let $A^h$ be the localization at $\mathfrak n^D$ of the integral closure $B^D$ of $A$ in $K^D_\text{sep}$. (Here it is easy to see that
$$ B^D= \{b \in B \ \vert \ \sigma(b) = b \text{ for all } \sigma \in D \}.) $$
Next, Milne shows that $A^h$ is Henselian. I understand the arguments used. But I do net got the argument why $A^h$ is even a Henselization.
Milne writes: To see that $A^h$ is the Henselization, one only has to show that it is a union of étale neighborhoods of $A$, but this is easy using (3.21).
Theorem 3.21: Let $X$ be a connected normal scheme, and let $K =R(X)$ the fraction field.
Let $L$ be a finite separable field extension of $K$, let $X'$ be the normalization of $X$ in $L$, and let $U$ be any open subscheme of $X'$ that is disjoint from
the support of $\Omega^1 _{X'/X}$ (sheaf of relative Kähler differentials). Then $U \to X$ is étale […]
Question: At the end of the paragraph is claimed that the quoted Theorem 3.21 helps to realize $A^h$ as union of étale neighbourhoods of $A$. How should (3.21) be applied here to reach the claimed result?
Idea: $\Omega^1 _{X'/X}$ has closed support. Therefore we could try to exhaust $A_h$ by union of localization $B^D_r$ for $r \in B^D- \mathfrak n^D$ with $D(r)$ disjoint from $\operatorname{Supp}(\Omega^1 _{X'/X})$.
But this comes with two problems:
Problem 1: all such $D(r)$ which should be étale over $A$ contain by choices of $r \in B^D- n^D$ the ‘point’ $n^D$, therefore we tacitly assume that $A^h$ is already étale over $ A$ in $\mathfrak n^D$, don't we? But why does this hold?
Problem 2: Are these also étale neighbourhoods of $A$? Compare with definition above: Note that these shold contain prime ideals $\mathfrak q$ lying over $\mathfrak m$ with residue fields isomorphic to $A/\mathfrak{m} =\kappa(\mathfrak q)$. I don't know how to control this condition.
Remark: I asked an almost identical question in MSE without getting an satisfying answer. Maybe this platform is more appropriate for this question.
