The Stacks project

Lemma 15.87.3. Let $K = (K_ n^\bullet )$ be an object of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$. There exists a canonical distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits K \to \prod \nolimits _ n K_ n^\bullet \to \prod \nolimits _ n K_ n^\bullet \to R\mathop{\mathrm{lim}}\nolimits K[1] \]

in $D(A)$. In other words, $R\mathop{\mathrm{lim}}\nolimits K$ is a derived limit of the inverse system $(K_ n^\bullet )$ of $D(A)$, see Derived Categories, Definition 13.34.1.

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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