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)