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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.