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 10.90.3. Let $R$ be a Noetherian ring and $A$ a set. Then $M = R^ A$ is a flat and Mittag-Leffler $R$-module.

Proof. Combining Lemma 10.89.5 and Proposition 10.89.6 we see that $M$ is flat over $R$. We show that $M$ satisfies the condition of Lemma 10.90.2. Let $F$ be a free finite $R$-module. If $F'$ is any submodule of $F$ then it is finitely presented since $R$ is Noetherian. So by Proposition 10.88.3 we have a commutative diagram

\[ \xymatrix{ F' \otimes _ R M \ar[r] \ar[d]^{\cong } & F \otimes _ R M \ar[d]^{\cong } \\ (F')^ A \ar[r] & F^ A } \]

by which we can identify the map $F' \otimes _ R M \to F \otimes _ R M$ with $(F')^ A \to F^ A$. Hence if $x \in F \otimes _ R M$ corresponds to $(x_\alpha ) \in F^ A$, then the submodule of $F'$ of $F$ generated by the $x_\alpha $ is the smallest submodule of $F$ such that $x \in F' \otimes _ R M$. $\square$


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