Corollary

*Let $\rho : A \to B$ be a homomorphism of noetherian rings making $B$ a flat $A$-module. If $A$ is regular and if $\kappa(\rho^{-1}(\mathfrak{n})) \otimes_{A} B$ is regular for every maximal ideal $\mathfrak{n}$ of $B$, the ring $B$ is regular.

For in fact, for every maximal ideal $\mathfrak{n}$ of $B$, the $A_{\rho^{-1}(\mathfrak{n})}$-module $B_n$ is flat (II, § 3, No. 4, Proposition 15), so that the ring $B_n$ is regular by Proposition 9.