# Structure of the sheaf $\frac{K(C)}{\mathcal{L}_D}$

I have a question about a sheaf and his stalks. Let $$\mathcal{A}$$ be $$\frac{K(C)}{\mathcal{L}_D}$$ ($$K(C)$$ is the function field of $$C$$ and $$\mathcal{L}_D$$ is the invertible sheaf associated to a divisor D) over a smooth complete curve $$C$$. $$\mathcal{A}$$ has the following propriety:

For every $$P \in C$$, for every $$U_{_P}$$ open neighborhood of P if $$s$$ is a section of $$\mathcal{A}(U_{_P})$$ exists an open neighborhood of $$P$$ $$U^{'}_{s,P}\subset U_{_P}$$ such that $$s_{\mid_{U^{'}_{s,P}-\{P\}}}=0$$.

Is possible to say that $$\mathcal{A}(C)$$ (global section of the sheaf) can be identified with $$\oplus_{P\in C} \mathcal{A}_P$$ (direct sum of the stalks)?

$$\textit{I've tried to prove the fact but i didn't succeed, maybe i wrong the approach:}$$

$$\textit{My idea is to create a sheaf \mathcal{A}_d(U):=\oplus_{P\in U}\mathcal{A}_P and then prove that this is isomorphic to \mathcal{A}}.$$

$$\textit{In this way i found two problems.}$$

$$\textit{First i was not able to prove that the gluing axiom on the sheaf \mathcal{A}_d holds}$$.

$$\textit{Second i am not sure that if i have an isomophism between the shaves}$$

$$\textit{i can use it to prove an isomorphism between the global section.}$$

(The only thing i have in mind is that it should came from the fact that the first chomology group of the sheaf $$\mathcal{A}_d$$ is trivial)

Someone could give me any suggestion please?