The Stacks project

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

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

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

  3. $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 $S$.

Proof. The assumption that $\mathfrak m$ is finitely generated implies that $R^\wedge $ is $\mathfrak mR^\wedge $-adically complete, that $\mathfrak mR^\wedge = \mathfrak m^\wedge $ and that $R^\wedge /\mathfrak m^ nR^\wedge = R/\mathfrak m^ n$, see Algebra, Lemma 10.96.3 and its proof. Similarly for $(S, \mathfrak n)$. Thus it is clear that diagrams as in Definition 15.37.1 for the cases (1), (2), and (3) are in 1-to-1 correspondence. $\square$

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