Theorem 4

Let A be a noetherian ring and B an A-algebra essentially of finite type. The following conditions are equivalent:

(i) the A-algebra B is formally smooth ; (ii) for every $q \in \mathrm{Spec}(B)$, the A-algebra $B_q$ is formally smooth (resp. formally smooth for the $qB_q$-adic topology) ; (iii) the A-module B is flat and, for every $p \in \mathrm{Spec}(A)$, the $\kappa(p)\text{-algebra} \ \kappa(p) \otimes_A B$ is absolutely regular ; (iv) the A-module B is flat and, for every regular A-algebra R, the ring $R \otimes_A B$ is regular ; (v) the A-module B is flat and the kernel of the homomorphism $\mu : B \otimes_A B \to B$ such that $\mu(x \otimes y) = xy$ is a completely secant ideal.

The equivalence of (i) and (ii) follows from cor. 2 of th. 3 (No. 9).

(i)⇒(v) : suppose the A-algebra B formally smooth. Let q be a prime ideal of B, and p its inverse image in A. The $A_p$-algebra $B_q$ is formally smooth (prop. 4, a) of No. 2), hence flat (lemma 5) ; consequently the A-module B is flat (II, § 3, No. 4, prop. 15). On the other hand, the ring $B \otimes_A B$ is noetherian (§ 6, No. 1, cor. of prop. 2), hence the ideal $\mathrm{Ker}\,\mu$ is completely secant by cor. 3 of th. 2 (No. 5).

(v)⇒(iii) : suppose condition (v) satisfied. Put $I = \mathrm{Ker}(\mu)$. Let $p \in \mathrm{Spec}(A)$. The mapping

is identified with the mapping

deduced from the multiplication of the $\kappa(p)\text{-algebra} \ \kappa(p) \otimes_A B$. The ideal $\mathrm{Ker}(\mu_p)$ is identified with $I(\kappa(p) \otimes_A (B \otimes_A B))$. It is completely secant since the A-module B is flat (§ 5, No. 6, prop. 6). Assertion (iii) then follows from prop. 8 of § 6, No. 5.

(iii)⇒(ii) : let q be a prime ideal of B, and p its inverse image in A. Under the hypotheses of (iii), the $A_p$-module $B_q$ is flat, and the $\kappa(p)\text{-algebra} \ \kappa(p) \otimes_{A_p} B_q$, which is identified with a ring of fractions of $\kappa(p) \otimes_A B$, is absolutely regular (§ 6, No. 4, prop. 6). It follows from lemma 5 that $B_q$ is formally smooth over $A_p$, hence over A (No. 2, prop. 3 and 4).

(iii)⇒(iv) : let us place ourselves under the hypotheses of (iii). Let R be a regular A-algebra. The R-module $R \otimes_A B$ is flat (I, § 2, No. 7, cor. 2 to prop. 8). Let r be a prime ideal of R and p its inverse image in A ; the ring $\kappa(r) \otimes_R (R \otimes_A B)$, which is identified with $\kappa(r) \otimes_{\kappa(p)} (\kappa(p) \otimes_A B)$, is regular (§ 6, No. 4, cor. 2 of prop. 7). The ring $R \otimes_A B$ is therefore regular (§ 4, No. 5, cor. of prop. 9).

(iv) ⇒ (iii) : let $p$ be a prime ideal of $A$ and let $k$ be an extension of $\kappa(p)$; under the hypotheses of (iv), the ring $k \otimes_{\kappa(p)} (\kappa(p) \otimes_A B)$, which is identified with $k \otimes_A B$, is regular, whence (iii).