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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?

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