The Stacks project

Lemma 10.160.7. Let $p > 0$ be a prime. Let $\Lambda $ be a Cohen ring with residue field of characteristic $p$. For every $n \geq 1$ the ring map

\[ \mathbf{Z}/p^ n\mathbf{Z} \to \Lambda /p^ n\Lambda \]

is formally smooth.

Proof. If $n = 1$, this follows from Proposition 10.158.9. For general $n$ we argue by induction on $n$. Namely, if $\mathbf{Z}/p^ n\mathbf{Z} \to \Lambda /p^ n\Lambda $ is formally smooth, then we can apply Lemma 10.138.12 to the ring map $\mathbf{Z}/p^{n + 1}\mathbf{Z} \to \Lambda /p^{n + 1}\Lambda $ and the ideal $I = (p^ n) \subset \mathbf{Z}/p^{n + 1}\mathbf{Z}$. $\square$

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