The Stacks project

Lemma 58.14.1. Let $I$ be a directed set. Let $X_ i$ be an inverse system of quasi-compact and quasi-separated schemes over $I$ with affine transition morphisms. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as in Limits, Section 32.2. Then there is an equivalence of categories

\[ \mathop{\mathrm{colim}}\nolimits \textit{FÉt}_{X_ i} = \textit{FÉt}_ X \]

If $X_ i$ is connected for all sufficiently large $i$ and $\overline{x}$ is a geometric point of $X$, then

\[ \pi _1(X, \overline{x}) = \mathop{\mathrm{lim}}\nolimits \pi _1(X_ i, \overline{x}) \]

Proof. The equivalence of categories follows from Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10. The second statement is formal given the statement on categories. $\square$

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