Proposition 3

Let $k$ be a ring.

a) Let $A$ and $B$ be linearly topologized $k$-algebras and $\rho : A \to B$ a continuous homomorphism of $k$-algebras. If $A$ is formally smooth over $k$ and $B$ formally smooth over $A$, then $B$ is formally smooth over $k$.

b) The product $k$-algebra of a finite family of linearly topologized formally smooth $k$-algebras is formally smooth.

c) Let $A$ be a linearly topologized $k$-algebra, and $\hat{A}$ the separated completion algebra of $A$; in order that $A$ be formally smooth over $k$, it is necessary and sufficient that this be so for $\hat{A}$.

Let C be a $k$-algebra, N a square-zero ideal of C, and $\pi : C \to C/N$ the canonical surjection. Equip C and C/N with the discrete topology.

a) Let $\psi : B \to C/N$ be a continuous homomorphism of $k$-algebras. Since A is formally smooth over $k$, there exists a continuous homomorphism of $k$-algebras $\tilde{\varphi} : A \to C$ such that $\pi \circ \tilde{\varphi} = \psi \circ \rho$.

Consider C and C/N as A-algebras by means of $\tilde{\varphi}$, so that $\psi$ is a homomorphism of A-algebras; since B is formally smooth over A, there exists a continuous homomorphism of A-algebras $\tilde{\psi} : B \to C$ such that $\pi \circ \tilde{\psi} = \psi$, whence a).

b) It is enough to prove that the product of two formally smooth $k$-algebras $A_1$ and $A_2$ is formally smooth. Let $\varphi : A_1 \times A_2 \to C/N$ be a continuous homomorphism of $k$-algebras. Put $e_1 = \varphi(1,0)$, $e_2 = \varphi(0,1)$, so that $e_1$ and $e_2$ are orthogonal idempotents in C/N. Let $\varphi_1 : A_1 \to (C/N)e_1$ and $\varphi_2 : A_2 \to (C/N)e_2$ denote the mappings defined by $\varphi_1(a_1) = \varphi(a_1,0)$ and $\varphi_2(a_2) = \varphi(0,a_2)$; these are continuous homomorphisms of $k$-algebras, and one has $\varphi(a_1, a_2) = \varphi_1(a_1) + \varphi_2(a_2)$ for every $(a_1, a_2) \in A_1 \times A_2$. There exists an idempotent element $\tilde{e}_1$ of C such that $\pi(\tilde{e}_1) = e_1$ (A, VIII, § 9, No. 4, Prop. 7); put $\tilde{e}_2 = 1 - \tilde{e}_1$, so that $\pi(\tilde{e}_2) = e_2$. For $i = 1, 2$, the homomorphism $C\tilde{e}_i \to (C/N)e_i$ induced by $\pi$ is surjective, with kernel $N\tilde{e}_i$; since the $k$-algebra $A_i$ is formally smooth, the homomorphism $\varphi_i$ admits a continuous lifting $\tilde{\varphi}_i$ to $C\tilde{e}_i$. The mapping $(a_1, a_2) \mapsto \tilde{\varphi}_1(a_1) + \tilde{\varphi}_2(a_2)$ is a continuous lifting of $\varphi$ to C.

c) Let $i : A \to \widehat{A}$ denote the canonical homomorphism. For every ring D, endowed with the discrete topology, the mapping which associates to a continuous homomorphism $f : \widehat{A} \to D$ the continuous homomorphism $f \circ i : A \to D$ is bijective. Assertion c) follows from this.

Assertion c) of the proposition applies in particular when the topology of $A$ is the J-adic topology, where J is an ideal of finite type; the closure $\widehat{J}$ of J in $\widehat{A}$ is then equal to $JA$ and the topology of $\widehat{A}$ is the $\widehat{J}$-adic topology (III, § 2, No. 12, Cor. 2 of Prop. 16). Consequently, it is equivalent to say that A is formally smooth for the J-adic topology or that its separated completion $\widehat{A}$ is formally smooth for the J-adic topology.