The Stacks project

Remark 13.34.4. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a complex of $\mathcal{A}$. Then $\tau _{\geq -n}K^\bullet $ is an inverse system of complexes and which in particular determines an inverse system in $D(\mathcal{A})$. Let us assume that $R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K^\bullet $ exists. Then the canonical maps $c_ n : K^\bullet \to \tau _{\geq -n}K^\bullet $ are compatible with the transition maps of our inverse system. By the defining distinguished triangle of Definition 13.34.1 and Lemma 13.4.2 we conclude there exists a morphism

in $D(\mathcal{A})$ such that the composition of $c$ with the projection $R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K^\bullet \to \tau _{\geq -m}K^\bullet $ is equal to $c_ m$. Now the morphism $c$ may not be unique, but we claim that whether or not $c$ is an isomorphism is independent of the choice of $c$ (and of our choice of the homotopy limit). Namely, for $i \in \mathbf{Z}$ and for $m > -i$ the composition

is the identity. Hence $H^ i(c)$ is an isomorphism if and only if the second map is an isomorphism. This is independent of $c$ and also independent of the choice of the homotopy limit (as any two choices are isomorphic).