Lemma 10.87.1. 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.
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