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 $K_ i \in \mathcal{D}$, $i = 1, \ldots , r$ 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 $\langle E \rangle$.