Lemma 15.87.4. With notation as in Lemma 15.87.3 the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences

of $A$-modules.

Lemma 15.87.4. With notation as in Lemma 15.87.3 the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits _ n H^{p - 1}(K_ n^\bullet ) \to H^ p(R\mathop{\mathrm{lim}}\nolimits K) \to \mathop{\mathrm{lim}}\nolimits _ n H^ p(K_ n^\bullet ) \to 0 \]

of $A$-modules.

**Proof.**
The proof is exactly the same as the proof of Lemma 15.86.7 using Lemma 15.87.1 in stead of Lemma 15.86.1.
$\square$

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