Lemma 4

Suppose that A is formally smooth over k for the J-adic topology. In order that $A/J$ be formally smooth over k, it is necessary and sufficient that the canonical mapping $\bar{d}$ have an A-linear retraction.

It is already known that if $A/J$ is formally smooth over $k$, the mapping $\bar{d}$ admits an A-linear retraction (No. 2, Remark 1). Conversely, suppose that $\bar{d}$ has an A-linear retraction. Let $\pi : A/J^2 \to A/J$ be the canonical surjection; by Prop. 2 of No. 1, there exists a ring homomorphism $h : A/J \to A/J^2$ such that $\pi \circ h = \mathrm{Id}_{A/J}$. Let C be a $k$-algebra, N an ideal of C which is square zero, and $\rho : C \to C/N$ the canonical surjection; endow C and C/N with the discrete topology. Let $u : A/J \to C/N$ be a continuous homomorphism of $k$-algebras. Since A is formally smooth over $k$ for the J-adic topology, there exists a homomorphism of $k$-algebras $v : A \to C$ making the diagram

commutative, where the vertical arrows represent the canonical surjections. One has $v(J) \subset \mathbf{N}$, hence $v(J^2) \subset \mathbf{N}^2 = \{0\}$, and $v$ defines, by passing to quotients, a homomorphism $\bar{v} : A/J^2 \to C$ which satisfies $\rho \circ \bar{v} = u \circ \pi$. Then $\bar{v} \circ h$ is a lifting of $u$ to $C$.