Lemma 5

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings. Suppose that $B$ is essentially of finite type over $A$. For the $A$-algebra $B$ to be formally smooth, it is necessary and sufficient that the $A$-module $B$ be flat and that the $\kappa_A$-algebra $\kappa_A \otimes_A B$ be absolutely regular.

There exists an integer $n \geqslant 0$, a prime ideal $q$ of $A[T_1, \ldots, T_n]$, and a surjective homomorphism $h$ of $A[T_1, \ldots, T_n]_q$ onto $B$. Let $C$ denote the local $A$-algebra $A[T_1, \ldots, T_n]_q$; it is formally smooth (No. 3, Example 2 and No. 2, Prop. 4, a)) and flat over $A$, and one can identify $B$ with the $A$-algebra $C/J$, where $J = \mathrm{Ker}(h)$.

Put $\overline{C} = \kappa_A \otimes_A C$ and $\overline{B} = \kappa_A \otimes_A B$. Suppose that $B$ is formally smooth over $A$. Then the $\kappa_A$-algebra $\overline{B}$ is formally smooth (No. 2, Prop. 4, b)), hence absolutely regular (No. 5, Cor. 2 of Th. 2). Moreover, since $\overline{C}/J\overline{C}$ can be identified with $\overline{B}$ and the $\kappa_A$-algebra $\overline{C}$ is formally smooth, the ideal $J\overline{C}$ of $\overline{C}$ is completely secant (No. 5, Th. 2). It then follows from § 5, No. 6, Prop. 6 that the $A$-module $B$ is flat.

Suppose conversely that $B$ is flat over $A$ and that the $\kappa_A$-algebra $\overline{B}$ is absolutely regular. Then the local $\kappa_A$-algebra $\overline{B}$ is formally smooth (Remark 1 of No. 9 with $k = k_0 = \kappa_A$). Put $\overline{J} = \kappa_A \otimes_A J$; since $B$ is a flat $A$-module, the canonical mapping $\overline{J} \to J\overline{C}$ is bijective and $\overline{B}$ can be identified with $\overline{C}/\overline{J}$. It follows (Remark 1 of No. 2) that the canonical mapping

is injective and admits a retraction. Now $\overline{J}/\overline{J}^2$ is identified with $\kappa_A \otimes_A J/J^2$, hence with $\overline{B} \otimes_B J/J^2$; on the other hand, the $\overline{C}$-module $\Omega_{\kappa_A}(\overline{C})$ is canonically isomorphic to $\overline{C} \otimes_C \Omega_A(C)$ (A, III, p. 136, prop. 20), hence $\overline{B} \otimes_{\overline{C}} \Omega_{\kappa_A}(\overline{C})$ is canonically isomorphic to $\overline{B} \otimes_C \Omega_A(C)$. Passing to the quotient by the maximal ideal of $\overline{B}$, one obtains an injective homomorphism

which is none other than $d_{\kappa_B}$. Thus B is formally smooth over A (th. 3).