Definition 1

One says that $A$ is formally smooth over $k$, or is a formally smooth $k$-algebra, if it satisfies the following condition: whatever the $k$-algebra $C$ and the square-zero ideal $N$ of $C$ may be, every continuous homomorphism from $A$ into the $k$-algebra $C/N$, endowed with the discrete topology, can be lifted to a continuous homomorphism from $A$ into the $k$-algebra $C$, endowed with the discrete topology.

Recall that a homomorphism from $A$ into a $k$-algebra endowed with the discrete topology is continuous if and only if its kernel is open.

Let $k$ be a ring, $A$ a $k$-algebra and $J$ an ideal of $A$. Endow $A$ with the $J$-adic topology. Let $C$ be a $k$-algebra, $N$ a square-zero ideal of $C$; endow $C$ and $C/N$ with the discrete topology. Let $\varphi : A \to C/N$ be a continuous homomorphism of algebras. Every lifting $\tilde{\varphi} : A \to C$ of $\varphi$ is continuous: in fact there exists an integer $n$ such that $\varphi(J^n)$ is zero, and one has $\tilde{\varphi}(J^n) \subset N$, whence $\tilde{\varphi}(J^{2n}) \subset N^2 = 0$. It follows in particular that, if $A$ is formally smooth for the $J$-adic topology, it is also formally smooth for the $J'$-adic topology for every ideal $J'$ containing $J$.

We shall say that a $k$-algebra $A$ is formally smooth if it is formally smooth when endowed with the discrete topology, that is to say, the $(0)$-adic topology; it is then formally smooth for the $J$-adic topology whatever the ideal $J$ of $A$ may be.