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Following question to the proof of Cor. 7.1.19 in Liu's "Algebraic Geometry" (p. 257):

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Why it is here already enough to show that every invertible sheaf $\mathcal{L}$ is isomorphic to a sub-$\mathcal{O}_X$-module of $\mathcal{O}_X$?, where $CaCl(X) \to Pic(X)$ is the image from Cartier Divisors modulo priciple Divisors to Picard group (for more detailed description see below):

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  • $\begingroup$ Because of point $(c)$ of the proposition 1.18 (the one you didn't reproduce here). It says that the image of $\mathcal{\rho}$ are the invertible sheaves contained in $\mathcal{K}_X$. $\endgroup$ – Roland Oct 8 '17 at 9:28
  • $\begingroup$ Sorry, I have attatched it now. So the argument is that every invertible sheaf which is a sub-$\mathcal{O}_X$-module can be embedded canonically in $\mathcal{K}_X$? $\endgroup$ – KarlPeter Oct 8 '17 at 10:42
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    $\begingroup$ What ?? The point (c) says that the image of $\rho$ are the locally free sheaves which are isomorphic (not necessarily canonically) to a sub-$\mathcal{O}_X$-module of $\mathcal{K}_X$. So if you prove that every locally free sheaf is isomorphic to a sub-$\mathcal{O}_X$-module of $\mathcal{K}_X$, you are done. $\endgroup$ – Roland Oct 8 '17 at 10:54

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