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The Stacks project

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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