Theorem 3

Let $k$ be a ring, $A$ a formally smooth $k$-algebra, and $J$ a finitely generated ideal of $A$; put $B = A/J$.

a) Let $\mathfrak{p}$ be a prime ideal of $B$ and let $q$ be the (prime) ideal of $A$ such that $\mathfrak{p} = q/J$. The following conditions are equivalent: (i) the $k$-algebra $B_{\mathfrak{p}}$ is formally smooth; (ii) there exists $f \in B - \mathfrak{p}$ such that the $k$-algebra $B_f$ is formally smooth; (iii) the $\kappa(\mathfrak{p})$-linear mapping

is injective; (iv) there exist an integer $m \geqslant 0$, elements $f_1, \ldots, f_m$ of $J$, whose images $(f_1)_q, \ldots, (f_m)_q$ generate the ideal $J_q$, and $k$-derivations $D_1, \ldots, D_m$ of $A$ such that $\det(D_j(f_i)) \notin q$.

b) The set of prime ideals $\mathfrak{p}$ of $B$ which satisfy the equivalent conditions of a) is open in $\mathrm{Spec}(B)$. In order that $B$ be formally smooth over $k$, it is necessary and sufficient that every prime (resp. maximal) ideal of $B$ satisfy these conditions.

c) Suppose that $A$ is noetherian. The conditions of a) are also equivalent to: (v) the $k$-algebra $B_{\mathfrak{p}}$ is formally smooth for the $\mathfrak{p}B_{\mathfrak{p}}$-adic topology. Moreover, under the conditions of (iv), the ideal $J_q$ is completely secant and the sequence $((f_1)_q, \ldots, (f_m)_q)$ is completely secant for $A_q$.

Put $M = J/J^2$ and $N = B \otimes_A \Omega_k(A)$. The $B$-module $M$ is of finite type, and the $B$-module $N$ is projective (No. 5, Corollary 3 of Theorem 2). For every multiplicative subset $S$ of $A$, the $k$-algebra $S^{-1}A$ is formally smooth (No. 2, Proposition 4, a)). By Lemma 4, conditions (i) and (ii) are therefore equivalent respectively to (i') the mapping $\bar{d}_{B_{\mathfrak{p}}} : M_{\mathfrak{p}} \to N_{\mathfrak{p}}$ has a $B_{\mathfrak{p}}$-linear retraction; (ii') there exists $f \in B - \mathfrak{p}$ such that the mapping $\bar{d}_{B_f} : M_f \to N_f$ has a $B_f$-linear retraction.

Prop. 9 of No. 8 applied to the ring $B$ and to the homomorphism $\bar{d} : M \to N$ implies the equivalence of conditions (i'), (ii'') and (iii), and also entails the assertions of b) (using again Lemma 4). Moreover, (iii) is equivalent to: (iii') the mapping $\kappa(q) \otimes_A J \to \kappa(q) \otimes_A \Omega_k(A)$ deduced from $d : J \to \Omega_k(A)$ is injective, whereas (iv) may be written: (iv') there exist an integer $m \geqslant 0$, elements $f_1, \ldots, f_m$ of $J$ whose images generate the ideal $J_q$ of $A_q$, and elements $y_1, \ldots, y_m$ of $\mathrm{Hom}_A(\Omega_k(A), A)$ such that $\det(<y_j, df_i>) \notin q$.

Since the $A$-module $\Omega_k(A)$ is projective (No. 5, Cor. 3 of Th. 2), Prop. 9 of No. 8, applied to the ring $A$ and to the homomorphism $d : J \to \Omega_k(A)$, yields the equivalence of (iii') and (iv').

Finally, suppose the ring $A$ noetherian. It is clear that (i) implies (v). Under assumption (v), Lemma 3 shows that the mapping

is injective, whence (iii).

Under the conditions of (iv), one has $m = [\kappa(q) \otimes_A J : \kappa(q)]$ (No. 9, Prop. 8). By Th. 2 of No. 5, the ideal $J_q$ is completely secant, and the sequence $((f_1)_q, \ldots, (f_m)_q)$ is completely secant for $A_q$ ($§ 1$, No. 3, Cor. 2 of Th. 1). This proves c).