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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)