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Let $Z=\{1/n:n\in\mathbb{Z}-\{0\}\}$. Now define the sheaf $\mathcal{J}_Z$ as $J_Z(U)=\{f:f\text{ is a holomorphic function on U with vanishing on }Z\}. $ Find $\text{Supp}(\mathcal{J}_Z).$

My attempt: for each $x\in\mathbb{C}$, if $x=0$, then by identity theorem, the only holomorphic function near zero & vanishing on $Z$ is the zero function, so $0\notin \text{Supp}(\mathcal{J}_Z)$. If $x\neq 0$, clearly the function $\sin(\pi/z)$ is a holomorphic function vanishing at $Z$, and defined everywhere but zero. Therefore, each stalk at $x\neq 0$ is nonzero, so $\text{Supp}(\mathcal{J}_Z)=\mathbb{C}-\{0\}.$

Am I correct? Please indicate if I have any mistakes.

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  • $\begingroup$ Yes you are correct. $\endgroup$ – Roland Mar 19 at 15:36

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