The Stacks project

Lemma 10.43.2. Elementary properties of geometrically reduced algebras. Let $k$ be a field. Let $S$ be a $k$-algebra.

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

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

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

  4. If $S$ is geometrically reduced over $k$, then any localization of $S$ is geometrically reduced over $k$.

Proof. Omitted. The second and third property follow from the fact that tensor product commutes with colimits. $\square$

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