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Suppose we have a scheme $S$ and an etale covering $\{ U_i \to S \}$. How much information about $\pi_1^{et} (S)$ can we recover from all the $\pi_1^{et} (U_i)$ ?

Are there special conditions we can put on $S$ to make this more tractable?

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    $\begingroup$ I think you can recover $\pi_1(S)$ from $\pi_1(U_i)$, $\pi_1(U_i\times_S U_j)$, $\pi_1(U_i\times_S U_j\times_S U_k)$ (roughly speaking, since we're missing base points). More precisely, one can recover the category of finite etale coverings of $S$ as the systems of finite etale coverings $V_i/U_i$ endowed with isomorphisms $f_{ij}\colon p_1^* V_i \simeq p_2^* V_j$ on $U_i\times_S V_j$, satisfying the cocycle condition $f_{ik} = f_{jk}f_{ij}$ on $U_i\times_S U_j\times_S U_k$. $\endgroup$ – Piotr Achinger Mar 19 '19 at 5:40
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    $\begingroup$ Otherwise, you can't say much, unfortunately. For example, for $S$ a smooth projective curve over $\mathbb{C}$ and $U_i\subseteq S$ an affine open, we have $\pi_1(U)$ free, but $\pi_1(S)$ is not free. $\endgroup$ – Piotr Achinger Mar 19 '19 at 5:46

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