Exercise 5, § 7

Let $A$ be a complete noetherian local ring and $B$ an essentially finite type $A$-algebra.

a) Now prove that the set of prime ideals $p$ of $A$ such that the ring $A_p$ is regular is open in $\mathrm{Spec}(A)$ (one will use Exercise 16 of VIII, § 5 to reduce to the case where $A$ is integral, then IX, § 2, No. 5, Theorem 3 and Exercise 17 of VIII, § 5).

b) Suppose that $A$ contains a field\footnote{This assumption is in fact superfluous, \emph{cf.} M. Nagata, \emph{On the Closedness of Singular Loci}, Publi. Mat. I.H.E.S. \textbf{2} (1959).}. Now prove that the set of prime ideals $q$ of $B$ such that the ring $B_q$ is regular is open in $\mathrm{Spec}(B)$ (one will use § 7, No. 9, Corollary 2 of Theorem 3 and IX, § 3, No. 3, Theorem 2).