Lemma 33.12.3. Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. The following are equivalent

$X$ is geometrically regular,

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

$X_{k'}$ is a regular 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 regular (see Algebra, Definition 10.166.2), and

there exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is geometrically regular over $k$.

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