In some progress of an exercise, I want to compute $\mathcal{Ext}^1(\mathcal{O}_L, \mathcal{O}_\mathbb{P^2})$. To achieve this, I started from the sequence $$ 0\rightarrow \mathcal{O}_{\mathbb{P^2}}(-1) \rightarrow \mathcal{O}_{\mathbb{P^2}} \rightarrow \mathcal{O}_L \rightarrow 0 $$
where $L$ is a line in the projective plane $\mathbb{P^2}$. Then taking the Hom sheaf funtor $\mathcal{Hom}(-,\mathcal{O}_\mathbb{P^2})$, it goes to $$ 0\rightarrow \mathcal{Hom}(\mathcal{O}_L,\mathcal{O}_\mathbb{P^2}) \rightarrow \mathcal{O}_\mathbb{P^2} \rightarrow \mathcal{O}_{\mathbb{P^2}}(1) \rightarrow \mathcal{Ext}^1(\mathcal{O}_L, \mathcal{O}_\mathbb{P^2}) \rightarrow 0 $$ Now $\mathcal{Hom}(\mathcal{O}_L, \mathcal{O}_{\mathbb{P^2}})=0$(Thanks to the comment), so I have an short sequence
$$ 0\rightarrow \mathcal{O}_\mathbb{P^2} \rightarrow \mathcal{O}_{\mathbb{P^2}}(1) \rightarrow \mathcal{Ext}^1(\mathcal{O}_L, \mathcal{O}_\mathbb{P^2}) \rightarrow 0$$ Next, recall the first basic short exact sequence with tensoring $\mathcal{O}_\mathbb{P^2}(2)$, I also have $$ 0\rightarrow \mathcal{O}_\mathbb{P^2}(1) \rightarrow \mathcal{O}_{\mathbb{P^2}}(2) \rightarrow \mathcal{O}_L(2) \rightarrow 0$$ Connecting those sequences \begin{eqnarray} 0&\rightarrow &\qquad\mathcal{O}_\mathbb{P^2} &\rightarrow &\quad\mathcal{O}_{\mathbb{P^2}}(1) &\rightarrow &\quad\mathcal{Ext}^1(\mathcal{O}_L, \mathcal{O}_\mathbb{P^2}) &\rightarrow &0\\\\ 0&\rightarrow &\quad\mathcal{O}_\mathbb{P^2}(1) &\rightarrow &\quad\mathcal{O}_{\mathbb{P^2}}(2) &\rightarrow &\quad\qquad\mathcal{O}_L(2) &\rightarrow &0 \end{eqnarray} and using snake lemma(Sorry, I can't write vertical down arrows), I got the third sequence $$ 0\rightarrow \mathcal{O}_L(1) \rightarrow \mathcal{O}_L(2) \rightarrow \mathcal{O}_p(2) \rightarrow 0$$ consequently, $$ \mathcal{Ext}^1(\mathcal{O}_L, \mathcal{O}_\mathbb{P^2})\cong\mathcal{O}_L(1). $$ Are those steps are correct?
Thanks for your help.