Lemma 15.51.9. Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$“$R$ is geometrically reduced over $k$”.

**Proof.**
Part (A) follows from the definition of geometrically reduced algebras (Algebra, Definition 10.43.1). Part (B) follows too: a ring is reduced if and only if all local rings are reduced. Part (C). This follows from Lemma 15.42.1. Part (D). This follows from Algebra, Lemma 10.164.2. Part (E). This follows from Algebra, Lemma 10.43.9.
$\square$

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