Is there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$

Question

I'd like to ask that if there a non-coherent subsheaf of $\mathscr{O}(-1)$ on $\mathbb{P}^1$.

I cannot think of an example.

Answer 1

Let $X=\mathbb{P}^1$ and $U$ be any non trivial open subset. For the sake of the example, let it be $U=\mathbb{P}^1\setminus\{\infty\}$ which is affine. Let $j:U\to \mathbb{P}^1$ the inclusion. Then $j_!j^{-1}\mathscr{O}_X$ (the extension by zero) will be a sub-$\mathscr{O}_X$-module of $\mathscr{O}_X$ which is not quasi-coherent.

To see this, note that you have a short exact sequences of sheaves : $$0\to j_!j^{-1}\mathscr{O}_X\to \mathscr{O}_X\to i_*\mathscr{O}_{\operatorname{Spec}\mathscr{O}_{X,\infty}}\to 0$$ where $i:\{\infty\}\to\mathbb{P}^1$ is the closed inclusion.

Now, take the sections on $V=\mathbb{P}^1\setminus\{0\}=\operatorname{Spec}k[\frac{1}{x}]$. You have a short exact sequence :

$$0\to \Gamma(V,j_!j^{-1}\mathscr{O}_X)\to k\left[\frac{1}{x}\right]\to \mathscr{O}_{X,\infty}$$

Since, the map $k[\frac{1}{x}]\to\mathscr{O}_{X,\infty}$ in injective, it follows that $\Gamma(V,j_!j^{-1}\mathscr{O}_X)=0$. Hence $j_!j^{-1}\mathscr{O}_X$ cannot be quasi-coherent (otherwise it would be zero which is not the case).

To get a non quasi-coherent of $\mathscr{O}_X(-1)$, just take the tensor product with $\mathscr{O}_X(-1)$.