Proposition 9

*Let $\rho : A \to B$ be a local homomorphism of noetherian local rings. The following conditions are equivalent:

(i) the ring $B$ is regular and the $\kappa_B$-linear mapping

is injective; (ii) the rings $B$ and $\kappa_A \otimes_A B$ are regular and the $A$-module $B$ is flat; (iii) the rings $A$ and $\kappa_A \otimes_A B$ are regular and the $A$-module $B$ is flat; (iv) the rings $A$ and $\kappa_A \otimes_A B$ are regular, and one has

One has $\dim(B) \leq \dim(A) + \dim(\kappa_A \otimes_A B)$ (VIII, § 3, No. 4, Corollary 1 of Proposition 7); the equivalence of (i) and (iv) therefore follows from Lemma 1. Under hypothesis (ii), the $A$-module $B$ is faithfully flat since one has $\mathfrak{m}_A B \subset \mathfrak{m}_B \neq B$ (I, § 3, No. 1, Proposition 1), which implies (iii) by Proposition 8. The implication (iii) $\Rightarrow$ (iv) follows from VIII, loc. cit.

It now remains for us to prove that when the equivalent conditions (i) and (iv) are satisfied, the $A$-module $B$ is flat. Let $x$ be a system of coordinates of $A$. Since $d\rho$ is injective, the image of this sequence under $\rho$ forms part of a system of coordinates of $B$. Thus the sequence $x$ is completely secant for $A$ and for $B$ (VIII, § 5, No. 3, Proposition 2), and generates the ideal $\mathfrak{m}_A$ of $A$. By Remark 3 of A, X, p. 159, the $A$-module $\mathrm{Tor}_1^A(\kappa_A, B)$ is isomorphic to $H_1(x, B)$, and therefore is zero; it follows that $B$ is flat over $A$ (III, § 5, No. 2, Theorem 1 and No. 4, Proposition 2).