The Stacks project

Lemma 15.86.2. Let

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

be a short exact sequence of inverse systems of abelian groups. Then there is an associated $6$ term exact sequence $0 \to \mathop{\mathrm{lim}}\nolimits A_ i \to \mathop{\mathrm{lim}}\nolimits B_ i \to \mathop{\mathrm{lim}}\nolimits C_ i \to R^1\mathop{\mathrm{lim}}\nolimits A_ i \to R^1\mathop{\mathrm{lim}}\nolimits B_ i \to R^1\mathop{\mathrm{lim}}\nolimits C_ i \to 0$.

Proof. Follows from the vanishing in Lemma 15.86.1. $\square$

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