The Stacks project

Lemma 13.42.4. Let $\mathcal{D}$ be a triangulated category. Let

\[ A_ n \to B_ n \to C_ n \to A_ n[1] \]

be an inverse system of distinguished triangles. If the system $C_ n$ is pro-zero (essentially constant with value $0$), then the maps $A_ n \to B_ n$ determine a pro-isomorphism between the pro-object $(A_ n)$ and the pro-object $(B_ n)$.

Proof. For any object $X$ of $\mathcal{D}$ consider the exact sequence

\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (C_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (B_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (A_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (C_ n[-1], X) \to \]

Exactness follows from Lemma 13.4.2 combined with Algebra, Lemma 10.8.8. By assumption the first and last term are zero. Hence the map $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (B_ n, X) \to \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (A_ n, X)$ is an isomorphism for all $X$. The lemma follows from this and Categories, Remark 4.22.7. $\square$

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