Lemma 10.48.5 (037V)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.48: Geometrically connected algebras
Lemma 10.48.5 (cite)

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

If $S$ is geometrically connected over $k$ so is every $k$ -subalgebra.
If all finitely generated $k$ -subalgebras of $S$ are geometrically connected, then $S$ is geometrically connected.
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. $\square$

Comments (1)

Comment #748 by Keenan Kidwell on June 27, 2014 at 18:34

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

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