The Stacks project

Proposition 13.37.6. Let $\mathcal{D}$ be a triangulated category with direct sums. Let $E$ be a compact object of $\mathcal{D}$. The following are equivalent

  1. $E$ is a classical generator for $\mathcal{D}_ c$ and $\mathcal{D}$ is compactly generated, and

  2. $E$ is a generator for $\mathcal{D}$.

Proof. If $E$ is a classical generator for $\mathcal{D}_ c$, then $\mathcal{D}_ c = \langle E \rangle $. It follows formally from the assumption that $\mathcal{D}$ is compactly generated and Lemma 13.36.4 that $E$ is a generator for $\mathcal{D}$.

The converse is more interesting. Assume that $E$ is a generator for $\mathcal{D}$. Let $X$ be a compact object of $\mathcal{D}$. Apply Lemma 13.37.3 with $I = \{ 1\} $ and $E_1 = E$ to write

\[ X = \text{hocolim} X_ n \]

as in the lemma. Since $X$ is compact we find that $X \to \text{hocolim} X_ n$ factors through $X_ n$ for some $n$ (Lemma 13.33.9). Thus $X$ is a direct summand of $X_ n$. By Lemma 13.37.4 we see that $X$ is an object of $\langle E \rangle $ and the lemma is proven. $\square$

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