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Lemma 12.31.3. Let

\[ 0 \to (A_ i) \to (B_ i) \to (C_ i) \to 0 \]

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

  1. 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.

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

  3. 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.

Proof. Nice exercise. See Algebra, Lemma 10.87.1 for part (3). $\square$

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