Lemma 10.47.6. Let $k$ be a field. Let $S$ be a $k$-algebra.

If $S$ is geometrically irreducible over $k$ so is every $k$-subalgebra.

If all finitely generated $k$-subalgebras of $S$ are geometrically irreducible, then $S$ is geometrically irreducible.

A directed colimit of geometrically irreducible $k$-algebras is geometrically irreducible.

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