The Stacks project

Lemma 13.42.5. Let $\mathcal{A}$ be an abelian category.

\[ A_ n \to B_ n \]

be an inverse system of maps of $D(\mathcal{A})$. Assume

  1. there exist integers $a \leq b$ such that $H^ i(A_ n) = 0$ and $H^ i(B_ n) = 0$ for $i \not\in [a, b]$, and

  2. the inverse system of maps $H^ i(A_ n) \to H^ i(B_ n)$ of $\mathcal{A}$ define an isomorphism of pro-objects of $\mathcal{A}$ for all $i \in \mathbf{Z}$.

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. We can inductively extend the maps $A_ n \to B_ n$ to an inverse system of distinguished triangles $A_ n \to B_ n \to C_ n \to A_ n[1]$ by axiom TR3. By Lemma 13.42.4 it suffices to prove that $C_ n$ is pro-zero. By Lemma 13.42.3 it suffices to show that $H^ p(C_ n)$ is pro-zero for each $p$. This follows from assumption (2) and the long exact sequences

\[ H^ p(A_ n) \xrightarrow {\alpha _ n} H^ p(B_ n) \xrightarrow {\beta _ n} H^ p(C_ n) \xrightarrow {\delta _ n} H^{p + 1}(A_ n) \xrightarrow {\epsilon _ n} H^{p + 1}(B_ n) \]

Namely, for every $n$ we can find an $m > n$ such that $\mathop{\mathrm{Im}}(\beta _ m)$ maps to zero in $H^ p(C_ n)$ because we may choose $m$ such that $H^ p(B_ m) \to H^ p(B_ n)$ factors through $\alpha _ n : H^ p(A_ n) \to H^ p(B_ n)$. For a similar reason we may then choose $k > m$ such that $\mathop{\mathrm{Im}}(\delta _ k)$ maps to zero in $H^{p + 1}(A_ m)$. Then $H^ p(C_ k) \to H^ p(C_ n)$ is zero because $H^ p(C_ k) \to H^ p(C_ m)$ maps into $\mathop{\mathrm{Ker}}(\delta _ m)$ and $H^ p(C_ m) \to H^ p(C_ n)$ annihilates $\mathop{\mathrm{Ker}}(\delta _ m) = \mathop{\mathrm{Im}}(\beta _ m)$. $\square$

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