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 10.100.8 (Critère de platitude par fibres: Nilpotent case). Let

\[ \xymatrix{ S \ar[rr] & & S' \\ & R \ar[lu] \ar[ru] } \]

be a commutative diagram in the category of rings. Let $I \subset R$ be a nilpotent ideal and $M$ an $S'$-module. Assume

  1. The module $M/IM$ is a flat $S/IS$-module.

  2. The module $M$ is a flat $R$-module.

Then $M$ is a flat $S$-module and $S_{\mathfrak q}$ is flat over $R$ for every $\mathfrak q \subset S$ such that $M \otimes _ S \kappa (\mathfrak q)$ is nonzero.

Proof. As $M$ is flat over $R$ tensoring with the short exact sequence $0 \to I \to R \to R/I \to 0$ gives a short exact sequence

\[ 0 \to I \otimes _ R M \to M \to M/IM \to 0. \]

Note that $I \otimes _ R M \to IS \otimes _ S M$ is surjective. Combined with the above this means both maps in

\[ I \otimes _ R M \to IS \otimes _ S M \to M \]

are injective. Hence $\text{Tor}_1^ S(IS, M) = 0$ (see Remark 10.74.9) and we conclude that $M$ is a flat $S$-module by Lemma 10.98.8. To finish we need to show that $S_{\mathfrak q}$ is flat over $R$ for any prime $\mathfrak q \subset S$ such that $M \otimes _ S \kappa (\mathfrak q)$ is nonzero. This follows from Lemma 10.38.15 and 10.38.10. $\square$


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