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

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

is exact.

## Comments (0)