Let $X$ be a smooth projective variety and $D$ be a divisor. I came across the notation
$\mathcal F \vert_D \in \langle \mathcal O_D \rangle$
where $\mathcal F \in D^b (X)$ and $\langle \mathcal O_D \rangle$ denotes the smallest triangulated subcategory of $D^b (X)$ generated by the sheaf $\mathcal O_D$.
I have trouble understanding how big or small this triangulated category is.
For example, if $X$ is a surface and $D$ is a smooth rational curve, then $\langle \mathcal O_D \rangle$ contains directs sums of (shifts of) $\mathcal O_D$, but I reckon "not much else".
Is this correct?
If not, what other complexes of sheaves can one find in $\langle \mathcal O_D \rangle$?