Lemma 76.36.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Then $X_{\overline{y}}$ is connected, if and only if for every étale neighbourhood $(V, \overline{v}) \to (Y, \overline{y})$ where $V$ is a scheme the base change $X_ V \to V$ has connected fibre $X_ v$.
Proof. Since the category of étale neighbourhoods of $\overline{y}$ is cofiltered and contains a cofinal collection of schemes (Properties of Spaces, Lemma 66.19.3) we may replace $Y$ by one of these neighbourhoods and assume that $Y$ is a scheme. Let $y \in Y$ be the point corresponding to $\overline{y}$. Then $X_ y$ is geometrically connected over $\kappa (y)$ if and only if $X_{\overline{y}}$ is connected and if and only if $(X_ y)_{k'}$ is connected for every finite separable extension $k'$ of $\kappa (y)$. See Spaces over Fields, Section 72.12 and especially Lemma 72.12.8. By More on Morphisms, Lemma 37.35.2 there exists an affine étale neighbourhood $(V, v) \to (Y, y)$ such that $\kappa (s) \subset \kappa (u)$ is identified with $\kappa (s) \subset k'$ any given finite separable extension. The lemma follows. $\square$
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