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