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*}

Proposition 10.88.2. Let $M$ be an $R$-module. The following are equivalent:

  1. $M$ is finitely generated.

  2. For every family $(Q_{\alpha })_{\alpha \in A}$ of $R$-modules, the canonical map $M \otimes _ R \left( \prod _{\alpha } Q_{\alpha } \right) \to \prod _{\alpha } (M \otimes _ R Q_{\alpha })$ is surjective.

  3. For every $R$-module $Q$ and every set $A$, the canonical map $M \otimes _ R Q^{A} \to (M \otimes _ R Q)^{A}$ is surjective.

  4. For every set $A$, the canonical map $M \otimes _ R R^{A} \to M^{A}$ is surjective.

Proof. First we prove (1) implies (2). Choose a surjection $R^ n \to M$ and consider the commutative diagram

\[ \xymatrix{ R^ n \otimes _ R (\prod _{\alpha } Q_{\alpha }) \ar[r]^{\cong } \ar[d] & \prod _{\alpha } (R^ n \otimes _ R Q_{\alpha }) \ar[d] \\ M \otimes _ R (\prod _{\alpha } Q_{\alpha }) \ar[r] & \prod _{\alpha } ( M \otimes _ R Q_{\alpha }). } \]

The top arrow is an isomorphism and the vertical arrows are surjections. We conclude that the bottom arrow is a surjection.

Obviously (2) implies (3) implies (4), so it remains to prove (4) implies (1). In fact for (1) to hold it suffices that the element $d = (x)_{x \in M}$ of $M^ M$ is in the image of the map $f: M \otimes _ R R^{M} \to M^ M$. In this case $d = \sum _{i = 1}^{n} f(x_ i \otimes a_ i)$ for some $x_ i \in M$ and $a_ i \in R^ M$. If for $x \in M$ we write $p_ x: M^ M \to M$ for the projection onto the $x$-th factor, then

\[ x = p_ x(d) = \sum \nolimits _{i = 1}^{n} p_ x(f(x_ i \otimes a_ i)) = \sum \nolimits _{i=1}^{n} p_ x(a_ i) x_ i. \]

Thus $x_1, \ldots , x_ n$ generate $M$. $\square$


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