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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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