Corollary

Let $B$ be a regular integral ring and let $A$ be a noetherian subring of $B$ such that $B$ is a finitely generated $A$-module. The following conditions are equivalent:

(i) $A$ is regular ; (ii) $B$ is a projective $A$-module ; (iii) $B$ is a flat $A$-module ; (iv) $B$ is a faithfully flat $A$-module.

(i) $\Rightarrow$ (ii) : this follows from the Cor. of Prop. 5 of No. 3. (ii) $\Rightarrow$ (iii) : this follows from I, § 3, No. 1, Prop. 1. (iii) $\Rightarrow$ (iv) : for every prime ideal $\mathfrak{p}$ of $A$, one has $\mathfrak{p}B \neq B$ (V, § 2, No. 1, Cor. 1 of Theorem 1). It is then enough to apply I, § 3, No. 1, Prop. 1. (iv) $\Rightarrow$ (i) : this follows from Prop. 8, b).

For every noetherian local ring $A$, let us denote by $\delta(A)$ the integer

Recall (VIII, § 5, No. 1) that $\delta(A)$ is always positive and that its vanishing characterises regular local rings.

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings; from $\rho$ one deduces a $\kappa_A$-linear homomorphism $\mathfrak{m}_A / \mathfrak{m}_A^2 \to \mathfrak{m}_B / \mathfrak{m}_B^2$, whence a $\kappa_B$-linear homomorphism