The Stacks project

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

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

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

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

Proof. Let $S' \subset S$ be a subalgebra. Then for any extension $k'/k$ the ring map $S' \otimes _ k k' \to S \otimes _ k k'$ is injective also. Hence (1) follows from Lemma 10.30.5 (and the fact that the image of an irreducible space under a continuous map is irreducible). The second and third property follow from the fact that tensor product commutes with colimits. $\square$

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