Lemma 13.36.6. Let $\mathcal{D}$ be a triangulated category which has a strong generator. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a strong generator.

Proof. Let $E'$ be an object of $\mathcal{D}$ such that $\mathcal{D} = \langle E' \rangle _ n$. Since $\mathcal{D} = \langle E \rangle$ we see that $E' \in \langle E \rangle _ m$ for some $m \geq 1$ by Lemma 13.36.2. Then $\langle E' \rangle _1 \subset \langle E \rangle _ m$ hence

$\mathcal{D} = \langle E' \rangle _ n = smd( \langle E' \rangle _1 \star \ldots \star \langle E' \rangle _1) \subset smd( \langle E \rangle _ m \star \ldots \star \langle E \rangle _ m) = \langle E \rangle _{nm}$

as desired. Here we used Lemma 13.36.1. $\square$

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