Lemma 3

Let K be an ideal of A containing J. Suppose that there exists an integer m such that $J \cap K^m$ is contained in JK (this condition is satisfied if A is noetherian). If $A/J$ is formally smooth over k for the K/J-adic topology, the mapping $\bar{d}_{A/K} : A/K \otimes_{A/J} J/J^2 \longrightarrow A/K \otimes_A \Omega_k(A)$ has an A-linear retraction.

Let C denote the $k$-algebra $A/(JK + K^m)$; the ideal N = $(J + K^m)/(JK + K^m)$ of C is square zero and the quotient ring C/N identifies with $A/(J + K^m)$. Endow C and C/N with the discrete topology, and A/J with the K/J-adic topology. The canonical homomorphism $A/J \to A/(J + K^m)$ is continuous; it therefore has a lifting $\varphi : A/J \to A/(JK + K^m)$.

The mapping $a \mapsto a1_C - \varphi(a1_{A/J})$ of A into N is then a $k$-derivation (No. 1, Prop. 1), hence is written $a \mapsto u(da)$ with $u \in \mathrm{Hom}_A(\Omega_k(A), N)$. But the assumption $J \cap K^m \subset JK$ implies $J \cap (JK + K^m) = JK$, so that the canonical mapping $\psi : J/JK \to N$ is bijective; there therefore exists $v \in \mathrm{Hom}_{A/K}(A/K \otimes_A \Omega_k(A), J/JK)$ such that, for a in A, one has $a1_C = \varphi(a1_{A/J}) + \psi(v(1 \otimes da))$. Taking a in J, one sees that $v(1 \otimes da)$ is equal to the class of a in J/JK. Since $A/K \otimes_{A/J} J/J^2$ identifies with $J/JK$, $v$ is the required retraction.

The fact that the condition on K is satisfied when the algebra A is noetherian follows from III, § 3, No. 1, Cor. 2 of Prop. 1.