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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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