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Let $D$ be a Cartier Divisor on a scheme $X$ represented by $\{(U_i, f_i)\}$ where $U_i$ is an open cover on $X$, $f_i \in \Gamma(U_i, \mathcal{K}^*)$ and such that $f_i/f_j \in \Gamma(U_i \cap U_j, \mathcal{O}^*)$.

We define the sheaf associated to $D$ denoted by $\mathcal{L}(D)$ to be the sub $\mathcal{O}_X$-module of $\mathcal{K}$ generated by $f_i^{-1}$ on $U_i$.

Why take the inverse here?

It seems that $D$ as described already is a line bundle via the map $\mathcal{O}_X \vert_{U_i} \to \mathcal{D} \vert_{U_i}$ defined as $1 \mapsto f_i$.

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