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Suppose that $N$ be a vector subbundle of $M_1 \oplus M_2 \oplus ... \oplus M_k$ where $M_1, M_2, ..., M_k$ are line bundles on a smooth algebraic curve $X$. Let $N_i$ be the image of $N$ under the projection $M_1 \oplus M_2 \oplus ... \oplus M_k \to M_i$. Is it true that $N=N_1 \oplus N_2 \oplus ... \oplus N_k$?

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    $\begingroup$ I don't think that's even true for vector spaces. $\endgroup$ Jun 29, 2019 at 16:23
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    $\begingroup$ Think about the line $x_1=x_2$ in $\Bbb R\oplus\Bbb R$. This is just the vector space case of the answer Spenser gave. Start simple!! $\endgroup$ Jun 29, 2019 at 16:37

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No. Take $M_1=M_2=X\times\mathbb{C}$ and $N=\{(x,u,v)\in M_1\oplus M_2=X\times\mathbb{C}^2:u=v\}$. Then, $N_1=M_1$ and $N_2=M_2$ so $N\neq N_1\oplus N_2$.

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  • $\begingroup$ Thank you very much! Do you think you could give an example when $M_1$ and $M_2$ are non-isomorphic line bundles? $\endgroup$
    – Yellow Pig
    Jun 29, 2019 at 16:48

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