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 10.81.14. Let $R$ be a ring and let $M \to M'$ be a map of $R$-modules. If $M'$ is flat, then $M \to M'$ is universally injective if and only if $M/IM \to M'/IM'$ is injective for every finitely generated ideal $I$ of $R$.

Proof. It suffices to show that $M \otimes _ R Q \to M' \otimes _ R Q$ is injective for every finite $R$-module $Q$, see Theorem 10.81.3. Then $Q$ has a finite filtration $0 = Q_0 \subset Q_1 \subset \ldots \subset Q_ n = Q$ by submodules whose subquotients are isomorphic to cyclic modules $R/I_ i$, see Lemma 10.5.4. Since $M'$ is flat, we obtain a filtration

\[ \xymatrix{ M \otimes Q_1 \ar[r] \ar[d] & M \otimes Q_2 \ar[r] \ar[d] & \ldots \ar[r] & M \otimes Q \ar[d] \\ M' \otimes Q_1 \ar@{^{(}->}[r] & M' \otimes Q_2 \ar@{^{(}->}[r] & \ldots \ar@{^{(}->}[r] & M' \otimes Q } \]

of $M' \otimes _ R Q$ by submodules $M' \otimes _ R Q_ i$ whose successive quotients are $M' \otimes _ R R/I_ i = M'/I_ iM'$. A simple induction argument shows that it suffices to check $M/I_ i M \to M'/I_ i M'$ is injective. Note that the collection of finitely generated ideals $I'_ i \subset I_ i$ is a directed set. Thus $M/I_ iM = \mathop{\mathrm{colim}}\nolimits M/I'_ iM$ is a filtered colimit, similarly for $M'$, the maps $M/I'_ iM \to M'/I'_ i M'$ are injective by assumption, and since filtered colimits are exact (Lemma 10.8.8) we conclude. $\square$


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