Lemma 10.165.6. Let k'/k be a separable algebraic field extension. Let A be an algebra over k'. Then A is geometrically normal over k if and only if it is geometrically normal over k'.
Proof. Let L/k be a finite purely inseparable field extension. Then L' = k' \otimes _ k L is a field (see material in Fields, Section 9.28) and A \otimes _ k L = A \otimes _{k'} L'. Hence if A is geometrically normal over k', then A is geometrically normal over k.
Assume A is geometrically normal over k. Let K/k' be a field extension. Then
K \otimes _{k'} A = (K \otimes _ k A) \otimes _{(k' \otimes _ k k')} k'
Since k' \otimes _ k k' \to k' is a localization by Lemma 10.43.8, we see that K \otimes _{k'} A is a localization of a normal ring, hence normal. \square
Comments (0)