Lemma 10.48.5. Let $k$ be a field. Let $S$ be a $k$-algebra.

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

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

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

Proof. This follows from the characterization of connectedness in terms of the nonexistence of nontrivial idempotents. The second and third property follow from the fact that tensor product commutes with colimits. $\square$

Comment #748 by Keenan Kidwell on

In (3), "irreducible" should be "connected."

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