Exercise 17, § 5

Let ρ : A → B be an injective homomorphism of noetherian integral rings making B a finitely generated A-module. a) Suppose that Reg(B) contains a nonempty open subset of Spec(B). Prove that Reg(A) contains a nonempty open subset of Spec(A). (Reduce to the case where B is flat over A by using II, § 5, No. 1, corollary to prop. 2, then to the case where B is regular by localizing with respect to a suitable element x ∈ A. Conclude using Exercise 14, a).) b) Suppose that Reg(A) contains a nonempty open subset of Spec(A) and that the field K’ of fractions of B is a separable extension of the field K of fractions of A. Prove that Reg(B) contains a nonempty open subset of Spec(B). (Let Z ∈ K’ be such that K’ = K(Z) (A, V, p. 39), and let P(X) ∈ K[X] be the minimal polynomial of Z. After replacing A by the ring $A_f$ for a suitable element $f$ of $A$, prove that one can reduce to proving the required property of $\mathrm{Spec}(B)$ in the following case: $\alpha) \ Z \in B$ and $A$ is regular ; $\beta)$ one has $P(X) \in A[X]$; $\gamma)$ one has $B = A[X]/(P(X))$ (II, § 5, No. 1, prop. 2); $\delta)$ $P'(Z)$ is invertible in $B$ (note that $P'(Z) \neq 0$ in $K'$ because $K'$ is separable over $K$. Use then II, § 5, No. 1, corollary to prop. 3). Prove that in this case, for every maximal ideal $m$ of $A$, $B/mB$ is an $(A/m)$-étale algebra (A, V, p. 32), hence regular. Conclude using Exercise 14, b).)