Section 72.13 (0DMT): Geometrically irreducible algebraic spaces

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72.13 Geometrically irreducible algebraic spaces

Spaces, Example 65.14.9 shows that it is best not to think about irreducible algebraic spaces in complete generality ¹. 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 72.13.1. 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 irreducible ² for any field extension $k'$ of $k$ .

Observe that $X_{k'}$ is a decent algebraic space (Decent Spaces, Lemma 68.6.5). Hence the topological space $|X_{k'}|$ is sober. Decent Spaces, Proposition 68.12.4.

[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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72.13 Geometrically irreducible algebraic spaces

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