Lemma 33.7.4. Let $k$ be a field. Let $X$, $Y$ be schemes over $k$. Assume $X$ is geometrically connected over $k$. Then the projection morphism

$p : X \times _ k Y \longrightarrow Y$

induces a bijection between connected components.

Proof. The scheme theoretic fibres of $p$ are connected, since they are base changes of the geometrically connected scheme $X$ by field extensions. Moreover the scheme theoretic fibres are homeomorphic to the set theoretic fibres, see Schemes, Lemma 26.18.5. By Morphisms, Lemma 29.23.4 the map $p$ is open. Thus we may apply Topology, Lemma 5.7.6 to conclude. $\square$

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