Cokernel of $\mathcal{O}_m\to\mathcal{O}_n$ on $\mathbb{P}^1$

Question

Consider the Serre twisted sheaf $\mathcal{O}_m$ and $\mathcal{O}_n$ with $m<n$ on $\mathbb{P}^1_k$ where $k$ is a field, as $\mathrm{Hom}(\mathcal{O}_m, \mathcal{O}_n)=\Gamma(\mathcal{O}_{n-m})$, there exist non-trivial maps $\mathcal{O}_m\to\mathcal{O}_n$.

I'd like to know how to consider the cokernel of some non-trivial map $\mathcal{O}_m\to\mathcal{O}_n$, is it a sheaf with finite support?

Thanks.

Answer 1

First, let's consider the special case $n=0$ and $m = -d<0$. In this case a non-trivial morphism $\mathcal{O}(-d) \to \mathcal{O}$ is given by a degree d homogeneous polynomial $f(x, y) \in \Gamma(\mathbb{P^1}, \mathcal{O}(d)) = k[x, y]_{(d)}$. Let us denote the closed subscheme $\{f = 0\} \subset \mathbb{P^1}$ by $Z$. There is a natural exact sequence $$ 0 \to \mathcal{O}(-d) \xrightarrow{f \cdot -} \mathcal{O} \to \mathcal{O}_Z \to 0$$ and therefore the cokernel is the structure sheaf of $Z$.

For general $n$ and $m$, the exact sequence above looks like $$ 0 \to \mathcal{O}(n-m) \xrightarrow{f \cdot -} \mathcal{O} \to \mathcal{O}_Z \to 0.$$ By tensoring $\mathcal{O}(m)$ there is another exact sequence $$ 0 \to \mathcal{O}(n) \xrightarrow{f \cdot -} \mathcal{O}(m) \to \mathcal{O}_Z \to 0$$ which illustrates that the cokernel is given by $\mathcal{O}_Z$.