The Stacks project

Lemma 37.53.3. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Then $X_ s$ is geometrically connected, if and only if for every étale neighbourhood $(U, u) \to (S, s)$ the base change $X_ U \to U$ has connected fibre $X_ u$.

Proof. If $X_ s$ is geometrically connected, then any base change of it is connected. On the other hand, suppose that $X_ s$ is not geometrically connected. Then by Varieties, Lemma 33.7.11 we see that $X_ s \times _{\mathop{\mathrm{Spec}}(\kappa (s))} \mathop{\mathrm{Spec}}(k)$ is disconnected for some finite separable field extension $k/\kappa (s)$. By Lemma 37.35.2 there exists an affine étale neighbourhood $(U, u) \to (S, s)$ such that $\kappa (u)/\kappa (s)$ is identified with $k/\kappa (s)$. In this case $X_ u$ is disconnected. $\square$

Comments (2)

Comment #7357 by Yijin Wang on

Typo in lemma 37.52.3: In the forth line 'we see that X_s×Spec(κ(s) Spec(k) is disconnected ' should be 'we see that X_s×Spec(κ(s)) Spec(k) is disconnected '

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  • 3 comment(s) on Section 37.53: Stein factorization

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