Exercise 14, § 5

Let ρ : A → B be a homomorphism of noetherian rings making B a faithfully flat A-module. (a) If B is regular, then A is regular (cf. p. 94, exerc. 2 and 3). (b) If A is regular and if, for every maximal ideal m of A, B/mB is regular, then B is regular. (By localizing reduce to the case where A and B are local and ρ is local. We then have dim(B) = dim(A) + dim(B/mB) (§ 3, No. 4, Cor. 1 to Prop. 7). Construct a regular sequence for B having dim(B) terms.)