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 10.136.12. Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal. Assume

  1. $I^2 = 0$,

  2. $R \to S$ is flat, and

  3. $R/I \to S/IS$ is formally smooth.

Then $R \to S$ is formally smooth.

Proof. Assume (1), (2) and (3). Let $P = R[\{ x_ t\} _{t \in T}] \to S$ be a surjection of $R$-algebras with kernel $J$. Thus $0 \to J \to P \to S \to 0$ is a short exact sequence of flat $R$-modules. This implies that $I \otimes _ R S = IS$, $I \otimes _ R P = IP$ and $I \otimes _ R J = IJ$ as well as $J \cap IP = IJ$. We will use throughout the proof that

\[ \Omega _{(S/IS)/(R/I)} = \Omega _{S/R} \otimes _ S (S/IS) = \Omega _{S/R} \otimes _ R R/I = \Omega _{S/R} / I\Omega _{S/R} \]

and similarly for $P$ (see Lemma 10.130.12). By Lemma 10.136.7 the sequence

10.136.12.1
\begin{equation} \label{algebra-equation-split} 0 \to J/(IJ + J^2) \to \Omega _{P/R} \otimes _ P S/IS \to \Omega _{S/R} \otimes _ S S/IS \to 0 \end{equation}

is split exact. Of course the middle term is $\bigoplus _{t \in T} S/IS \text{d}x_ t$. Choose a splitting $\sigma : \Omega _{P/R} \otimes _ P S/IS \to J/(IJ + J^2)$. For each $t \in T$ choose an element $f_ t \in J$ which maps to $\sigma (\text{d}x_ t)$ in $J/(IJ + J^2)$. This determines a unique $S$-module map

\[ \tilde\sigma : \Omega _{P/R} \otimes _ R S = \bigoplus S\text{d}x_ t \longrightarrow J/J^2 \]

with the property that $\tilde\sigma (\text{d}x_ t) = f_ t$. As $\sigma $ is a section to $\text{d}$ the difference

\[ \Delta = \text{id}_{J/J^2} - \tilde\sigma \circ \text{d} \]

is a self map $J/J^2 \to J/J^2$ whose image is contained in $(IJ + J^2)/J^2$. In particular $\Delta ((IJ + J^2)/J^2) = 0$ because $I^2 = 0$. This means that $\Delta $ factors as

\[ J/J^2 \to J/(IJ + J^2) \xrightarrow {\overline{\Delta }} (IJ + J^2)/J^2 \to J/J^2 \]

where $\overline{\Delta }$ is a $S/IS$-module map. Using again that the sequence (10.136.12.1) is split, we can find a $S/IS$-module map $\overline{\delta } : \Omega _{P/R} \otimes _ P S/IS \to (IJ + J^2)/J^2$ such that $\overline{\delta } \circ d$ is equal to $\overline{\Delta }$. In the same manner as above the map $\overline{\delta }$ determines an $S$-module map $\delta : \Omega _{P/R} \otimes _ P S \to J/J^2$. After replacing $\tilde\sigma $ by $\tilde\sigma + \delta $ a simple computation shows that $\Delta = 0$. In other words $\tilde\sigma $ is a section of $J/J^2 \to \Omega _{P/R} \otimes _ P S$. By Lemma 10.136.7 we conclude that $R \to S$ is formally smooth. $\square$


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