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 10.86.1. Let $R$ be a ring. Let $0 \to K_ i \to L_ i \to M_ i \to 0$ be short exact sequences of $R$-modules, $i \geq 1$ which fit into maps of short exact sequences

\[ \xymatrix{ 0 \ar[r] & K_ i \ar[r] & L_ i \ar[r] & M_ i \ar[r] & 0 \\ 0 \ar[r] & K_{i + 1} \ar[r] \ar[u] & L_{i + 1} \ar[r] \ar[u] & M_{i + 1} \ar[r] \ar[u] & 0} \]

If for every $i$ there exists a $c = c(i) \geq i$ such that $\mathop{\mathrm{Im}}(K_ c \to K_ i) = \mathop{\mathrm{Im}}(K_ j \to K_ i)$ for all $j \geq c$, then the sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits K_ i \to \mathop{\mathrm{lim}}\nolimits L_ i \to \mathop{\mathrm{lim}}\nolimits M_ i \to 0 \]

is exact.

Proof. This is a special case of the more general Lemma 10.85.4. $\square$


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