Proposition 4

Let k be a ring, A and B $k$-algebras, J an ideal of A, K an ideal of B.

a) Let S be a multiplicative subset of A and T a subset of k whose image in A is contained in S. If A is formally smooth over k for the J-adic topology, S $^1A$ is formally smooth over $T^{-1}k$ for the $S^{-1}J$-adic topology.

b) Let $k'$ be a $k$-algebra. If $A$ is formally smooth over $k$ for the $J$-adic topology, the $k'$-algebra $A_{(k')}$ is formally smooth over $k'$ for the $JA_{(k')}$-adic topology.

c) Let $I$ denote the ideal of $A \otimes_k B$ generated by the images of $J \otimes_k B$ and $A \otimes_k K$. If $A$ and $B$ are formally smooth over $k$ for the $J$-adic and $K$-adic topologies respectively, the $k$-algebra $A \otimes_k B$ is formally smooth for the $I$-adic topology.

a) Under the hypotheses of a), let $C$ be a $T^{-1}k$-algebra, and $N$ an ideal of square zero of $C$; equip $C$ and $C/N$ with the discrete topology, and let $\pi : C \to C/N$ denote the canonical surjection. Let $\varphi : S^{-1}A \longrightarrow C/N$ be a homomorphism of $T^{-1}k$-algebras, continuous for the $S^{-1}J$-adic topology. Let $i$ denote the canonical homomorphism of $A$ into $S^{-1}A$. The mapping $\varphi \circ i$ is a homomorphism of $k$-algebras, continuous for the $J$-adic topology, and therefore admits a lifting $\tilde{\varphi}_0 : A \to C$. The elements of $\tilde{\varphi}_0(S)$ are invertible modulo $N$, hence invertible since $N$ is square zero. It follows that there exists a ring homomorphism $\tilde{\varphi} : S^{-1}A \to C$ such that $\tilde{\varphi} \circ i = \tilde{\varphi}_0$ (II, § 2, No. 1, Proposition 1); by Corollary 3 to Proposition 2 of loc. cit., $\tilde{\varphi}$ is $T^{-1}k$-linear. We have $\pi \circ \tilde{\varphi} \circ i = \varphi \circ i$, whence $\pi \circ \tilde{\varphi} = \varphi$ (loc. cit., Proposition 1), so that $\tilde{\varphi}$ is a lifting of $\varphi$.

b) Let us place ourselves under the hypotheses of b). Let $C$ be a $k'$-algebra, $N$ a square-zero ideal of $C$; let us endow $C$ and $C/N$ with the discrete topology. Let $\varphi : A_{(k')} \longrightarrow C/N$ be a homomorphism of $k'$-algebras, continuous for the $JA_{(k')}$-adic topology. Let $i : A \to A_{(k')}$ denote the canonical homomorphism. The mapping $\varphi \circ i$ is a homomorphism of $k$-algebras from $A$ into $C/N$, continuous for the $J$-adic topology; if $A$ is formally smooth over $k$ for the $J$-adic topology, $\varphi \circ i$ admits a lifting $\tilde{\psi} : A \to C$. The homomorphism of $k'$-algebras $\tilde{\varphi} : A_{(k')} \longrightarrow C$ deduced from $\tilde{\psi}$ is a lifting of $\varphi$.

c) Let us place ourselves under the hypotheses of c). The $B$-algebra $A \otimes_k B$ is formally smooth for the $J(A \otimes_k B)$-adic topology by b), hence for the $I$-adic topology; moreover the canonical homomorphism $B \to A \otimes_k B$ is continuous when $B$ is endowed with the $K$-adic topology and $A \otimes_k B$ with the $I$-adic topology. The assertion c) therefore results from prop. 3, a).