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- Jun 20, 2014

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Let $(X,\mathscr{O})$ be a ringed space. Suppose $\mathscr{F}$ is an invertible sheaf over $\mathscr{O}$. That is, $\mathscr{F}$ is a rank one locally free module over $\mathscr{O}$. Prove that there is an isomorphism between the tensor sheaf $\mathscr{F}\otimes_\mathcal{O}\check{\mathscr{F}}$ and structure sheaf $\mathscr{O}$, where $\check{\mathscr{F}} = \operatorname{Hom}_X(\mathscr{F},\mathscr{O})$.

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