Lemma 33.10.3. Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent

$X$ is geometrically normal,

$X_{k'}$ is a normal scheme for every field extension $k'/k$,

$X_{k'}$ is a normal scheme for every finitely generated field extension $k'/k$,

$X_{k'}$ is a normal scheme for every finite purely inseparable field extension $k'/k$,

for every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is geometrically normal (see Algebra, Definition 10.165.2), and

$X_{k^{perf}}$ is a normal scheme.

## Comments (0)