The Stacks project

Lemma 15.51.10. Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$“$R$ is geometrically normal over $k$”.

Proof. Part (A) follows from the definition of geometrically normal algebras (Algebra, Definition 10.165.2). Part (B) follows too: a ring is normal if and only if all of its local rings are normal. Part (C). This follows from Lemma 15.42.2. Part (D). This follows from Algebra, Lemma 10.164.3. Part (E). This follows from Algebra, Lemma 10.165.6. $\square$

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