Lemma 12.31.3. Let

be a short exact sequence of inverse systems of abelian groups.

In any case the sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits _ i A_ i \to \mathop{\mathrm{lim}}\nolimits _ i B_ i \to \mathop{\mathrm{lim}}\nolimits _ i C_ i \]is exact.

If $(B_ i)$ is ML, then also $(C_ i)$ is ML.

If $(A_ i)$ is ML, then

\[ 0 \to \mathop{\mathrm{lim}}\nolimits _ i A_ i \to \mathop{\mathrm{lim}}\nolimits _ i B_ i \to \mathop{\mathrm{lim}}\nolimits _ i C_ i \to 0 \]is exact.

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