Lemma 15.37.4. Let $(R, \mathfrak m)$ and $(S, \mathfrak n)$ be rings endowed with finitely generated ideals. Endow $R$ and $S$ with the $\mathfrak m$-adic and $\mathfrak n$-adic topologies. Let $R \to S$ be a homomorphism of topological rings. The following are equivalent

$R \to S$ is formally smooth for the $\mathfrak n$-adic topology,

$R \to S^\wedge $ is formally smooth for the $\mathfrak n^\wedge $-adic topology,

$R^\wedge \to S^\wedge $ is formally smooth for the $\mathfrak n^\wedge $-adic topology.

Here $R^\wedge $ and $S^\wedge $ are the $\mathfrak m$-adic and $\mathfrak n$-adic completions of $R$ and $S$.

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