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 10.98.7 (Local criterion for flatness). Let $R \to S$ be a local homomorphism of local Noetherian rings. Let $\mathfrak m$ be the maximal ideal of $R$, and let $\kappa = R/\mathfrak m$. Let $M$ be a finite $S$-module. If $\text{Tor}_1^ R(\kappa , M) = 0$, then $M$ is flat over $R$.

Proof. Let $I \subset R$ be an ideal. By Lemma 10.38.5 it suffices to show that $I \otimes _ R M \to M$ is injective. By Remark 10.74.9 we see that this kernel is equal to $\text{Tor}_1^ R(M, R/I)$. By Lemma 10.98.6 we see that $J \otimes _ R M \to M$ is injective for all ideals of finite colength.

Choose $n >> 0$ and consider the following short exact sequence

\[ 0 \to I \cap \mathfrak m^ n \to I \oplus \mathfrak m^ n \to I + \mathfrak m^ n \to 0 \]

This is a sub sequence of the short exact sequence $0 \to R \to R^{\oplus 2} \to R \to 0$. Thus we get the diagram

\[ \xymatrix{ (I\cap \mathfrak m^ n) \otimes _ R M \ar[r] \ar[d] & I \otimes _ R M \oplus \mathfrak m^ n \otimes _ R M \ar[r] \ar[d] & (I + \mathfrak m^ n) \otimes _ R M \ar[d] \\ M \ar[r] & M \oplus M \ar[r] & M } \]

Note that $I + \mathfrak m^ n$ and $\mathfrak m^ n$ are ideals of finite colength. Thus a diagram chase shows that $\mathop{\mathrm{Ker}}((I \cap \mathfrak m^ n)\otimes _ R M \to M) \to \mathop{\mathrm{Ker}}(I \otimes _ R M \to M)$ is surjective. We conclude in particular that $K = \mathop{\mathrm{Ker}}(I \otimes _ R M \to M)$ is contained in the image of $(I \cap \mathfrak m^ n) \otimes _ R M$ in $I \otimes _ R M$. By Artin-Rees, Lemma 10.50.2 we see that $K$ is contained in $\mathfrak m^{n-c}(I \otimes _ R M)$ for some $c > 0$ and all $n >> 0$. Since $I \otimes _ R M$ is a finite $S$-module (!) and since $S$ is Noetherian, we see that this implies $K = 0$. Namely, the above implies $K$ maps to zero in the $\mathfrak mS$-adic completion of $I \otimes _ R M$. But the map from $S$ to its $\mathfrak mS$-adic completion is faithfully flat by Lemma 10.96.3. Hence $K = 0$, as desired. $\square$


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