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 10.88.10. If $M = \bigoplus _{i \in I} M_ i$ is a direct sum of $R$-modules, then $M$ is Mittag-Leffler if and only if each $M_ i$ is Mittag-Leffler.

Proof. The “only if” direction follows from Lemma 10.88.7 (1) and the fact that a split short exact sequence is universally exact. The converse follows from Lemma 10.88.9 but we can also argue it directly as follows. First note that if $I$ is finite then this follows from Lemma 10.88.7 (2). For general $I$, if all $M_ i$ are Mittag-Leffler then we prove the same of $M$ by verifying condition (1) of Proposition 10.87.6. Let $f: P \to M$ be a map from a finitely presented module $P$. Then $f$ factors as $P \xrightarrow {f'} \bigoplus _{i' \in I'} M_{i'} \hookrightarrow \bigoplus _{i \in I} M_ i$ for some finite subset $I'$ of $I$. By the finite case $\bigoplus _{i' \in I'} M_{i'}$ is Mittag-Leffler and hence there exists a finitely presented module $Q$ and a map $g: P \to Q$ such that $g$ and $f'$ dominate each other. Then also $g$ and $f$ dominate each other. $\square$


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