Proposition 2

Endow the $k$-module $A/J \oplus J/J^2$ with the structure of a $k$-algebra defined in the above example. The mappings $v \mapsto h_v$ and $h \mapsto \psi_h$ induce bijections between the following sets :

(i) the set of $A/J$-linear retractions $v$ of $\bar{d}$ ; (ii) the set of homomorphisms of $k$-algebras $h : A/J \to A/J^2$ such that $\pi \circ h = \mathrm{Id}_{A/J}$ ; (iii) the set of isomorphisms of $k$-algebras $\psi : A/J \oplus J/J^2 \longrightarrow A/J^2$ such that $\pi \circ \psi = \mathrm{pr}_1$ and $\psi(0, z) = z$ for $z \in J/J^2$.

Apply Prop. 1 with $C = A/J^2$ and $N = J/J^2$. Let $\varphi : A \to A/J$ be the canonical surjection ; the homomorphism $\rho$ is a lifting of $\varphi$ to $A/J^2$. The $A$-module $\mathrm{Hom}_{A/J}((A/J) \otimes_A \Omega_k(A), J/J^2)$ is identified with $\mathrm{Hom}_A(\Omega_k(A), J/J^2)$ ; after Prop. 1, the mapping $v \mapsto H_v$ is a bijection of this set onto the set of liftings of $\varphi$ to $A/J^2$. For $x \in J$, one has $1 \otimes dx = \bar{d}(\rho(x))$ ; for $H_v$ to vanish on J, it is necessary and sufficient that $v \circ \bar{d}$ be the identity mapping of $J/J^2$. This proves that the mapping $v \mapsto h_v$ induces a bijection between the first two sets described in the statement.

The mapping $h \mapsto \psi_h$ is a bijection of the set of $k$-linear homomorphisms of $A/J$ into $A/J^2$ onto the set of $k$-linear homomorphisms $\psi : A/J \oplus J/J^2 \longrightarrow A/J^2$ such that $\psi(0, z) = z$ for $z \in J/J^2$ ; moreover, for $\pi \circ \psi_h = \mathrm{pr}_1$ to hold, it is necessary and sufficient that $\pi \circ h = \mathrm{Id}_{A/J}$, that is to say $z \equiv h(\pi(z)) \pmod{J/J^2}$ for every $z \in A/J^2$. Suppose these conditions satisfied. For $h$ to be a ring homomorphism, it is necessary and sufficient that the same be true of $\psi_h$ ; moreover, the homomorphism $\psi_h$ is bijective : the inverse mapping associates to an element $z$ of $A/J^2$ the pair $(\pi(z), z - h(\pi(z)))$. This proves that the mapping $h \mapsto \psi_h$ induces a bijection between the last two sets described in the statement.