Lemma 33.8.4 (038F)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.8: Geometrically irreducible schemes
Lemma 33.8.4 (cite)

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Lemma 33.8.4. Let $k$ be a field. Let $X$ , $Y$ be schemes over $k$ . Assume $X$ is geometrically irreducible over $k$ . Then the projection morphism

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

induces a bijection between irreducible components.

Proof. First, note that the scheme theoretic fibres of $p$ are irreducible, since they are base changes of the geometrically irreducible 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 5.8.15 to conclude. $\square$

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