I have some problems to prove the exercise 4.13 (on page 39) from J.S. Milne's Etale Cohomology (the "book", not the online accessible script!):
Let $A$ be a Henselian local ring and $k=A/m$ it's residue field. We consider a smooth $A$-scheme $X$ with canonical map $f: X \to A$.
Problem The exercise is to show that the map
$$\operatorname{Hom}(\operatorname{Spec}(A), X)=X(A) \to X(k) = \operatorname{Hom}(\operatorname{Spec}(k), X)$$
between $A$-and $k$-valued points induced by $A \to k$ is surjective. or in other words that each map $\varphi: \operatorname{Spec}(k) \to X$ obtains a lift in $X(A)$.
This is what I tried:
let $x = \varphi(\{*\})$ the unique image of $\varphi$. since $f$ is smooth, there exist an open subscheme $U \subset X$ containing $x$ and an open $V \subset \operatorname{Spec}(A)$ with $f(U) \subset V$ such that the restriction of $f$ to $U$ factorizes as
$$U \xrightarrow{\text{g}} \mathbb{A}^n_V \xrightarrow{\text{h}} V$$
with $g$ étale and $h$ canonical. (this was application of Prop. 3.24(b) following the hint). Assume after restricting if neccessary that $U:=\operatorname{Spec}(B)$ and $V:=\operatorname{Spec}(R)$ are affine. since $V$ is open in $\operatorname{Spec}(A)$, we can assume that $R= A_s$ (i.e. a localization of $A$ at a $a \in A$).
The problem translates to comm. algebra:
abusing notation we have an etale ring map $g:A_s[X_1,\ldots, X_n] \to B$ and $\varphi: B \to k$, which we want to lift to $\bar{\varphi}: B \to A$.
Problems:
(1) In genral localizations of Henselian ring are not Henselian, thus $A_s$ is not Henselian in general, thus we cannot at this point apply Henselian lifting theorem to lift $A_s[X_1,\ldots, X_n] \to k$ to $A$. We need an argument that we can choose $s \in A$ such that $A_s=A$.
(2) assume we solved problem (1) and have a lift $A_s[X_1,\ldots, X_n] \to A$. can we show that it factorizes through $B$? which characterization of étaleness of $g$ could at this point do it's job?
Could anybody help me how to solve this problem?
Update #1: I think I have solved (1): $f \circ \varphi$ maps $\{*\}$ to the unique closed point $x_{\mathfrak{m}}$ of $\operatorname{Spec}(A)$ and every open $V \subset \operatorname{Spec}(A)$, which contains $x_{\mathfrak{m}}$ is already $\operatorname{Spec}(A)$, since $A$ local. therefore we can assume $V=\operatorname{Spec}(A)$.
What do we know about etale $g:A[X_1,\ldots, X_n] \to B$ and Hensel lifts? Is there a criterion which allows to lift zeros of more then one polynomial simulaneously?
Locally $f$ factors through $\mathbb{A}^n_A$ (by smoothness), and so we may assume this globally. The element of $X(k)$ gives an element of $\mathbb{A}^n(k)$, which (obviously) lifts to an $A$-morphism $Spec(A)\to \mathbb{A}^n_A$. Form the fibered product of this morphism with the morphism $X\to \mathbb{A}^n_A$ to get a scheme etale over $A$, and apply I 4.2(d) of Milne's book. [I think you were only missing the last step.]