Lemma 15.40.2 (Regular is a local property). Let $R \to \Lambda $ be a ring map with $\Lambda $ Noetherian. The following are equivalent

$R \to \Lambda $ is regular,

$R_\mathfrak p \to \Lambda _\mathfrak q$ is regular for all $\mathfrak q \subset \Lambda $ lying over $\mathfrak p \subset R$, and

$R_\mathfrak m \to \Lambda _{\mathfrak m'}$ is regular for all maximal ideals $\mathfrak m' \subset \Lambda $ lying over $\mathfrak m$ in $R$.

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