The Stacks project

Lemma 15.86.8. With notation as in Lemma 15.86.7 the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences

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

Proof. The long exact sequence of the distinguished triangle is

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

The map in the middle has kernel $\mathop{\mathrm{lim}}\nolimits _ n H^ p(K_ n^\bullet )$ by its explicit description given in the lemma. The cokernel of this map is $R^1\mathop{\mathrm{lim}}\nolimits _ n H^ p(K_ n^\bullet )$ by Lemma 15.86.1. $\square$

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