Lemma 72.12.4 (0A12)—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 72.12.4 (cite)

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Lemma 72.12.4. Let $k'/k$ be an extension of fields. Let $X$ be an algebraic space over $k$. Assume $k$ separably algebraically closed. Then the morphism $X_{k'} \to X$ induces a bijection of connected components. In particular, $X$ is geometrically connected over $k$ if and only if $X$ is connected.

Proof. Since $k$ is separably algebraically closed we see that $k'$ is geometrically connected over $k$, see Algebra, Lemma 10.48.4. Hence $Z = \mathop{\mathrm{Spec}}(k')$ is geometrically connected over $k$ by Varieties, Lemma 33.7.5. Since $X_{k'} = Z \times _ k X$ the result is a special case of Lemma 72.12.3. $\square$

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