The Stacks project

Lemma 13.42.3. Let $\mathcal{A}$ be an abelian category. Let $A_ n$ be an inverse system of objects of $D(\mathcal{A})$. Assume

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

  2. the inverse systems $H^ i(A_ n)$ of $\mathcal{A}$ are essentially constant for all $i \in \mathbf{Z}$.

Then $A_ n$ is an essentially constant system of $D(\mathcal{A})$ whose value $A$ satisfies that $H^ i(A)$ is the value of the constant system $H^ i(A_ n)$ for each $i \in \mathbf{Z}$.

Proof. By Remark 13.12.4 we obtain an inverse system of distinguished triangles

\[ \tau _{\leq a}A_ n \to A_ n \to \tau _{\geq a + 1}A_ n \to (\tau _{\leq a}A_ n)[1] \]

Of course we have $\tau _{\leq a}A_ n = H^ a(A_ n)[-a]$ in $D(\mathcal{A})$. Thus by assumption these form an essentially constant system. By induction on $b - a$ we find that the inverse system $\tau _{\geq a + 1}A_ n$ is essentially constant, say with value $A'$. By Lemma 13.42.2 we find that $A_ n$ is an essentially constant system. We omit the proof of the statement on cohomologies (hint: use the final part of Lemma 13.42.2). $\square$

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