The Stacks project

Lemma 13.35.8. Let $\mathcal{T}$ be a triangulated category. Let $H : \mathcal{T} \to \mathcal{A}$ be a homological functor to an abelian category $\mathcal{A}$. Let $a \leq b$ and $\mathcal{E} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{T})$ be a subset such that $H^ i(E) = 0$ for $E \in \mathcal{E}$ and $i \not\in [a, b]$. Then for $X \in smd(add(\mathcal{E}[-m, m])^{\star n})$ we have $H^ i(X) = 0$ for $i \not\in [-m + na, m + nb]$.

Proof. Omitted. Pleasant exercise in the definitions. $\square$

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