Exercise 7, § 7

Let $k$ be a ring, $\rho : A \to B$ a homomorphism of $k$-algebras, $J$ an ideal of $B$. We say that $B$, endowed with the $J$-adic topology, is \emph{formally smooth over} $A$ \emph{relative to} $k$ if for every $A$-algebra $C$ (which is endowed with the discrete topology) and every square zero ideal $N$ of $C$, every continuous homomorphism of $A$-algebras $B \to C/N$, which lifts to a continuous homomorphism of $A$-algebras from $B$ into $C$, also lifts to a continuous homomorphism of $A$-algebras from $B$ into $C$.

a) Now prove that the following conditions are equivalent :

(i) $B$ is formally smooth over $A$ relative to $k$ for the $J$-adic topology ; (ii) for every $k$-derivation $D$ of $A$ into a $B$-module $E$ annihilated by a power of $J$, there exists a $k$-derivation $\tilde{D} : B \to E$ such that $\tilde{D} \circ \rho = D$ ; (iii) for every integer $n \geqslant 0$, the canonical mapping $u_n : \Omega_k(A) \otimes_A (B/J^n) \longrightarrow \Omega_k(B) \otimes_B (B/J^n)$ is injective and its image is a direct factor.

b) Suppose that $B$ is formally smooth over $k$ for the $J$-adic topology. Prove that in order that $B$ be formally smooth over $A$, it is necessary and sufficient that the canonical homomorphism $u_1 : \Omega_k(A) \otimes_A (B/J) \to \Omega_k(B) \otimes_B (B/J)$ be injective and that its image be a direct factor (use a) and Exercise 6 to prove that a retraction of $u_1$ lifts to a retraction of $u_n$ for every $n$).