MHB Can an invertible sheaf be isomorphic to the structure sheaf?

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Here is another chance to solve a sheaf problem!

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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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No one answered this week's problem. You can read my solution below.

Spoiler

Let $x\in X$. Then there is an isomorphism $(\mathscr{F}\otimes_{\mathscr{O}} \check{\mathscr{F}})_x \approx \mathscr{F}_x \otimes_{\mathscr{O}_x} \check{\mathscr{F}}_x$. Since $\mathscr{F}$ is invertible over $\mathscr{O}$, $\mathscr{F}_x \approx \mathscr{O}_x$. Hence $\check{\mathscr{F}}_x \approx \operatorname{Hom}_{\mathscr{O}_x}(\mathscr{F}_x,\mathscr{O}_x) \approx \operatorname{Hom}_{\mathscr{O}_x}(\mathscr{O}_x,\mathscr{O}_x) \approx \mathscr{O}_x$, and so $\mathscr{F}_x \otimes_{\mathscr{O}_x} \check{\mathscr{F}}_x \approx \mathscr{O}_x \otimes_{\mathscr{O}_x} \mathscr{O}_x \approx \mathscr{O}_x$. Therefore, $(\mathscr{F} \otimes_\mathscr{O}\check{\mathscr{F}})_x \approx \mathscr{O}_x$. Since $x$ was arbitrary, the tensor sheaf $\mathscr{F}\otimes_{\mathscr{O}} \check{\mathscr{F}}$ is isomorphic to $\mathscr{O}$.