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let $E$ be a vector bundle of rank $r$ on $X\times \mathbb{P}^{1}$ where $X$ is a smooth projective curve. Assume now that $E|_{F_{p}} \cong \mathcal{O}_{\mathbb{P}^{1}}^{r}$ for any p-fiber where $p: X\times \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}$. It should be true that $E \cong p^{*}p_{*}E$. How to show that? Thanks

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The canonical homomorphism $\varphi : p^*p_*E\rightarrow E$ induces on each fiber $F_z$ $(z\in\mathbb{P}^1)$ the evaluation map $H^0(F_z, E_{|F_z})\otimes \mathscr{O}_{F_z}\rightarrow E_{|F_z}$. By your hypothesis this is an isomorphism, so $\varphi $ is an isomorphism.

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