Lemma 33.8.16 (054S)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.8: Geometrically irreducible schemes
Lemma 33.8.16 (cite)

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Lemma 33.8.16. Let $X$ be a scheme over the field $k$ . Assume $X$ has finitely many irreducible components which are all geometrically irreducible. Then $X$ has finitely many connected components each of which is geometrically connected.

Proof. This is clear because a connected component is a union of irreducible components. Details omitted. $\square$

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