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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09SR. Beware of the difference between the letter 'O' and the digit '0'.