The Stacks project

Remark 13.36.7. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. Let $T$ be a property of objects of $\mathcal{D}$. Suppose that

  1. if $K_ i \in D(A)$, $i = 1, \ldots , r$ with $T(K_ i)$ for $i = 1, \ldots , r$, then $T(\bigoplus K_ i)$,

  2. if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object,

  3. if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and

  4. $T(E[n])$ holds for all $n$.

Then $T$ holds for all objects of $\langle E \rangle $.


Comments (1)

Comment #8551 by Long Liu on

In Remark 0ATH (1), 'D(A)' should be 'D'.


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Your email address will not be published.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 0ATH. Beware of the difference between the letter 'O' and the digit '0'.