Lemma 72.12.2 (0A10)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.12: Geometrically connected algebraic spaces
Lemma 72.12.2 (cite)

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Lemma 72.12.2. Let $X$ be an algebraic space over the field $k$ . Let $k'/k$ be a field extension. Then $X$ is geometrically connected over $k$ if and only if $X_{k'}$ is geometrically connected over $k'$ .

Proof. If $X$ is geometrically connected over $k$ , then it is clear that $X_{k'}$ is geometrically connected over $k'$ . For the converse, note that for any field extension $k''/k$ there exists a common field extension $k'''/k'$ and $k'''/k'$ . As the morphism $X_{k'''} \to X_{k''}$ is surjective (as a base change of a surjective morphism between spectra of fields) we see that the connectedness of $X_{k'''}$ implies the connectedness of $X_{k''}$ . Thus if $X_{k'}$ is geometrically connected over $k'$ then $X$ is geometrically connected over $k$ . $\square$

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