Lemma 33.8.3 (020J)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.8: Geometrically irreducible schemes
Lemma 33.8.3 (cite)

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Lemma 33.8.3. Let $X$ be a scheme over a separably closed field $k$ . If $X$ is irreducible, then $X_ K$ is irreducible for any field extension $K/k$ . I.e., $X$ is geometrically irreducible over $k$ .

Proof. Use Properties, Lemma 28.3.3 and Algebra, Lemma 10.47.2. $\square$

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