The Stacks project

Lemma 58.10.6. Let $(A, \mathfrak m)$ be a local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $. Let $U^{sh}$ be the punctured spectrum of the strict henselization $A^{sh}$ of $A$. Assume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence

\[ \pi _1(U^{sh}, \overline{u}) \to \pi _1(U, \overline{u}) \to \pi _1(X, \overline{u}) \to 1 \]

is exact in the sense of Lemma 58.4.3 part (1).

Proof. The map $\pi _1(U) \to \pi _1(X)$ is surjective by Lemmas 58.10.2 and 58.4.1.

Write $X^{sh} = \mathop{\mathrm{Spec}}(A^{sh})$. Let $Y \to X$ be a finite étale morphism. Then $Y^{sh} = Y \times _ X X^{sh} \to X^{sh}$ is a finite étale morphism. Since $A^{sh}$ is strictly henselian we see that $Y^{sh}$ is isomorphic to a disjoint union of copies of $X^{sh}$. Thus the same is true for $Y \times _ X U^{sh}$. It follows that the composition $\pi _1(U^{sh}) \to \pi _1(U) \to \pi _1(X)$ is trivial, see Lemma 58.4.2.

To finish the proof, it suffices according to Lemma 58.4.3 to show the following: Given a finite étale morphism $V \to U$ such that $V \times _ U U^{sh}$ is a disjoint union of copies of $U^{sh}$, we can find a finite étale morphism $Y \to X$ with $V \cong Y \times _ X U$ over $U$. The assumption implies that there exists a finite étale morphism $Y^{sh} \to X^{sh}$ and an isomorphism $V \times _ U U^{sh} \cong Y^{sh} \times _{X^{sh}} U^{sh}$. Consider the following diagram

\[ \xymatrix{ U \ar[d] & U^{sh} \ar[d] \ar[l] & U^{sh} \times _ U U^{sh} \ar[d] \ar@<1ex>[l] \ar@<-1ex>[l] & U^{sh} \times _ U U^{sh} \times _ U U^{sh} \ar[d] \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] \\ X & X^{sh} \ar[l] & X^{sh} \times _ X X^{sh} \ar@<1ex>[l] \ar@<-1ex>[l] & X^{sh} \times _ X X^{sh} \times _ X X^{sh} \ar@<1ex>[l] \ar[l] \ar@<-1ex>[l] } \]

Since $U \subset X$ is quasi-compact by assumption, all the downward arrows are quasi-compact open immersions. Let $\xi \in X^{sh} \times _ X X^{sh}$ be a point not in $U^{sh} \times _ U U^{sh}$. Then $\xi $ lies over the closed point $x^{sh}$ of $X^{sh}$. Consider the local ring homomorphism

\[ A^{sh} = \mathcal{O}_{X^{sh}, x^{sh}} \to \mathcal{O}_{X^{sh} \times _ X X^{sh}, \xi } \]

determined by the first projection $X^{sh} \times _ X X^{sh}$. This is a filtered colimit of local homomorphisms which are localizations étale ring maps. Since $A^{sh}$ is strictly henselian, we conclude that it is an isomorphism. Since this holds for every $\xi $ in the complement it follows there are no specializations among these points and hence every such $\xi $ is a closed point (you can also prove this directly). As the local ring at $\xi $ is isomorphic to $A^{sh}$, it is strictly henselian and has connected punctured spectrum. Similarly for points $\xi $ of $X^{sh} \times _ X X^{sh} \times _ X X^{sh}$ not in $U^{sh} \times _ U U^{sh} \times _ U U^{sh}$. It follows from Lemma 58.10.4 that pullback along the vertical arrows induce fully faithful functors on the categories of finite étale schemes. Thus the canonical descent datum on $V \times _ U U^{sh}$ relative to the fpqc covering $\{ U^{sh} \to U\} $ translates into a descent datum for $Y^{sh}$ relative to the fpqc covering $\{ X^{sh} \to X\} $. Since $Y^{sh} \to X^{sh}$ is finite hence affine, this descent datum is effective (Descent, Lemma 35.37.1). Thus we get an affine morphism $Y \to X$ and an isomorphism $Y \times _ X X^{sh} \to Y^{sh}$ compatible with descent data. By fully faithfulness of descent data (as in Descent, Lemma 35.35.11) we get an isomorphism $V \to U \times _ X Y$. Finally, $Y \to X$ is finite étale as $Y^{sh} \to X^{sh}$ is, see Descent, Lemmas 35.23.29 and 35.23.23. $\square$

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