Lemma 15.86.13. Let

be a short exact sequence of inverse systems of abelian groups. If $(A_ i)$ and $(C_ i)$ are ML, then so is $(B_ i)$.

Lemma 15.86.13. Let

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

be a short exact sequence of inverse systems of abelian groups. If $(A_ i)$ and $(C_ i)$ are ML, then so is $(B_ i)$.

**Proof.**
This follows from Lemma 15.86.12, the fact that taking infinite direct sums is exact, and the long exact sequence of cohomology associated to $R\mathop{\mathrm{lim}}\nolimits $.
$\square$

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