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

1. $R \to \Lambda$ is regular,

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

3. $R_\mathfrak m \to \Lambda _{\mathfrak m'}$ 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 10.109.7 and a ring map is flat if and only if all the induced maps of local rings are flat, see Algebra, Lemma 10.38.19. $\square$

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