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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