Lemma 15.36.2. Let $\varphi : R \to S$ be a ring map.

If $R \to S$ is formally smooth in the sense of Algebra, Definition 10.136.1, then $R \to S$ is formally smooth for any linear topology on $R$ and any pre-adic topology on $S$ such that $R \to S$ is continuous.

Let $\mathfrak n \subset S$ and $\mathfrak m \subset R$ ideals such that $\varphi $ is continuous for the $\mathfrak m$-adic topology on $R$ and the $\mathfrak n$-adic topology on $S$. Then the following are equivalent

$\varphi $ is formally smooth for the $\mathfrak m$-adic topology on $R$ and the $\mathfrak n$-adic topology on $S$, and

$\varphi $ is formally smooth for the discrete topology on $R$ and the $\mathfrak n$-adic topology on $S$.

## Comments (0)