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I am following Vakil's notes on Foundations of algebraic geometry. We have just introduced twisting sheaves ${\cal O}(n)$ on $\mathbb P^1_k$ by gluing the structure sheaves on standard open subsets with a non-standard map.

Later in the text, we take a rational function $s \in {\cal O}_X(U)$ and associate to it a rational section of ${\cal O}(n)$, but the procedure on how this is done is not explained. It is not clear to me how to get a section of $s$, since we constructed ${\cal O}(n)$ by gluing and there seems to be some choice related to which chart of ${\cal O}(n)$ one chooses to interpret the function $s$. As ${\cal O}(n)$ is non-trivial, this choice will produce different rational functions.

Example: How can we interpret $1/x_0$ as a section of ${\cal O}(1)$? Or of ${\cal O}(-1)$? We can think of $1/x_0$ as defined on an open subset of $D(x_0)$ as well as an open subset of $D(x_1)$. If we start with $D(x_0)$, then it gives a regular function $1/x_0$ there, and this function corresponds to $x_1^2$ on $D(x_1)$ if we are working with ${\cal O}(1)$, and to $1$ on $D(x_1)$ if we are working with ${\cal O}(-1)$.

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