We consider $\mathbb{P}^1$ and the semiorthogonal decomposition $<\mathcal{O},\mathcal{O}(1)>=D^b(\mathbb{P}^1)$. Let $x$ be a closed point and $k(x)$ the corresponding skyscraper sheaf. Every object $\mathcal{F}$ in $D^b(\mathbb{P}^1)$ is sitting in a triangle:
$B\rightarrow \mathcal{F}\rightarrow A\rightarrow B[1]$, for $B$ in $<\mathcal{O}(1)>$ and $A$ in $<\mathcal{O}>$.
Now my question: Is it possible to describe explicitely the objects $A$ and $B$ if $\mathcal{F}=k(x)$? And is it possible to say anything about the support of the objects $A$ and $B$ in this case?