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70.13 Geometrically irreducible algebraic spaces

Spaces, Example 63.14.9 shows that it is best not to think about irreducible algebraic spaces in complete generality1. For decent (for example quasi-separated) algebraic spaces this kind of disaster doesn't happen. Thus we make the following definition only under the assumption that our algebraic space is decent.

Definition 70.13.1. Let $k$ be a field. Let $X$ be a decent algebraic space over $k$. We say $X$ is geometrically irreducible if the topological space $|X_{k'}|$ is irreducible2 for any field extension $k'$ of $k$.

Observe that $X_{k'}$ is a decent algebraic space (Decent Spaces, Lemma 66.6.5). Hence the topological space $|X_{k'}|$ is sober. Decent Spaces, Proposition 66.12.4.

[1] To be sure, if we say “the algebraic space $X$ is irreducible”, we probably mean to say “the topological space $|X|$ is irreducible”.
[2] An irreducible space is nonempty.

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