Let $X$ be the projective line over a field $k$ of positive characteristic. I am trying to prove that don't exist an exact sequence of coherent sheaves over $X$ $$0 \to \mathcal{O} \oplus \mathcal{O} \to \mathcal{O}(1) \oplus \mathcal{O}(1) \to \mathcal{K}_P \to 0$$ when $\mathcal{K}_P$ is the skyscraper sheaf of a degre $2$ point $P$.
Does anyone have a suggestion of how to do it? Or a counterexample?
I am trying to compute an example when $k= \mathbb{F}_2$ and $P$ correspond to the ideal $x^2+xy+y^2$ (Or, in the affine open set $\mathbb{A}^1_x$, to the point $\alpha$ such that $\alpha$ is a root of $x^2+x+1 \in \mathbb{F}_2[x]$).
Consider the morphism $\mathcal{O} \oplus \mathcal{O} \to \mathcal{O}(1) \oplus \mathcal{O}(1)$ given by the matrix $$ \begin{pmatrix} x & y \\ y & x + y \end{pmatrix}. $$ Its cokernel has length $2$ and is killed by the determinant of the matrix (which is equal to $x^2 + xy + y^2$), hence it is isomorphic to the structure sheaf of a degree $2$ point.