Exercise:
Let $X$ be a nonsingular curve over an algebraically closed field $k$.
(c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$(means that its stalk at the generic point has dimension $r$ as a $K(X)$-vector space), show that there is a divisor $D$ on $X$ and an exact sequence $0 \rightarrow \mathscr{L}(D)^{\oplus r}\to \mathscr {F} \rightarrow \mathscr J \rightarrow 0$, where $ \mathscr J$ is a torsion sheaf.(means that its stalk at the generic point is $0$. )
Next is the idea of an answer I have read:
To construct the injective morphism, take a basis for the $K(X)$-vector space $\mathscr{F}_{\xi}$, and find a suitable $\mathscr{L}(D)$ such that this basis gives global sections of $\mathscr{L}(D) \otimes \mathscr{F}$. This defines a morphism $\mathcal{O}_{X}^{\oplus n} \rightarrow \mathscr{L}(D) \otimes \mathscr{F}$ which we show to be injective, and then tensor everything with $\mathscr{L}(D)^{-1}$. Since locally free sheaf are flat at stalks, we are done. But I have troubles in conducting it(How to find such a proper $D$ and get the global sections?).
My efforts:
Take an finite open affine cover $U_i=\operatorname{Spec} A_i$ of $X$, then we can assume $\mathscr F|_{U_i}=\tilde {M_i}$. Pick a basis $e_1,\cdots,e_r$ of $\mathscr F_\xi$, and suppose $e_j=\frac{m_{ij}}{a_i}$ where $m_{ij}\in M_i,a_i\in A_i$.(here we can assume all $e_j$ has the same denominator by doing some elementary multiplications)
Denote $V_i=D(a_i)\subset U_i$. Then $V_i$ may not be a cover of $X$, but we can modify it. For each $x\in (\cup V_i)^c$ we can pick an index $i$ such that $U_i$ contains $x$. Now let $Z_i$ be all such $x$ that is arranged in $U_i$. Set $W_i=V_i\cup Z_i$(this is an open cover, since $X$ is a curve). Since $Z_i$ are disjoint from each other, we can easily show $U_i\cap U_j=V_i\cap V_j$.
Now we know that $a_i$ and $a_j$ are both invertible in $W_i\cap W_j$, we can use all $a_i$s to glue a cartier divisor $D$. $\mathscr L(D)$ is locally generated by $a_i^{-1}$. We consider the tensor product $\mathscr G:=\mathscr L(D)\otimes \mathscr F$. We know that $m_{ij}\otimes \frac{1}{a_i}\in \mathscr G(W_i)$. I want to glue $\{m_{ij}\otimes \frac{1}{a_i}\}_i$ together to get a global section, but I think they may not be compatible in $W_i\cap W_{i'}$.
Could you help me modify my solution? Other ways are also welcome!