Lemma 58.10.8. Let $X$ be a scheme. Let $x_1, \ldots , x_ n \in X$ be a finite number of closed points such that
$U = X \setminus \{ x_1, \ldots , x_ n\} $ is connected and is a retrocompact open of $X$, and
for each $i$ the punctured spectrum $U_ i^{sh}$ of the strict henselization of $\mathcal{O}_{X, x_ i}$ is connected.
Then the map $\pi _1(U) \to \pi _1(X)$ is surjective and the kernel is the smallest closed normal subgroup of $\pi _1(U)$ containing the image of $\pi _1(U_ i^{sh}) \to \pi _1(U)$ for $i = 1, \ldots , n$.
Proof.
Surjectivity follows from Lemmas 58.10.4 and 58.4.1. We can consider the sequence of maps
\[ \pi _1(U) \to \ldots \to \pi _1(X \setminus \{ x_1, x_2\} ) \to \pi _1(X \setminus \{ x_1\} ) \to \pi _1(X) \]
A group theory argument then shows it suffices to prove the statement on the kernel in the case $n = 1$ (details omitted). Write $x = x_1$, $U^{sh} = U_1^{sh}$, set $A = \mathcal{O}_{X, x}$, and let $A^{sh}$ be the strict henselization. Consider the diagram
\[ \xymatrix{ U \ar[d] & \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} \ar[l] \ar[d] & U^{sh} \ar[d] \ar[l] \\ X & \mathop{\mathrm{Spec}}(A) \ar[l] & \mathop{\mathrm{Spec}}(A^{sh}) \ar[l] } \]
By Lemma 58.4.3 we have to show finite étale morphisms $V \to U$ which pull back to trivial coverings of $U^{sh}$ extend to finite étale schemes over $X$. By Lemma 58.10.6 we know the corresponding statement for finite étale schemes over the punctured spectrum of $A$. However, by Limits, Lemma 32.20.1 schemes of finite presentation over $X$ are the same thing as schemes of finite presentation over $U$ and $A$ glued over the punctured spectrum of $A$. This finishes the proof.
$\square$
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