The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.154.7. 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 10.152.9. 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 10.136.12 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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