Lemma 15.41.2 (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 $R$.
Proof. This is true because a Noetherian ring is regular if and only if all the local rings are regular local rings, see Algebra, Definition 10.110.7 and a ring map is flat if and only if all the induced maps of local rings are flat, see Algebra, Lemma 10.39.18. $\square$
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