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> [!definition] Definition. ([[line bundle associated to a Cartier divisor]])
> Let $X$ be [[scheme]] and $\mathcal{K}_{X}$ the [[sheaf of rational functions]] on $X$. Let $D$ be a [[Cartier divisor]] on $X$; suppose $D$ is [[quotient sheaf|represented by]] local sections $\{ (U_{i}, f_{i}) \}$.
>
> Define the **line bundle associated to $D$**, $\mathcal{O}_{X}(D)$, to be the [[subsheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] of $\mathcal{K}_{X}$ [[subsheaf of modules generated by local sections|generated by]] the local sections $f_{i} ^{-1} \in \Gamma(U_{i}, \mathcal{K}_{X}^{*} )$. In other words, $\mathcal{O}_{X}(D)$ is the [[sheaf]] [[sheafification|associated to]] the [[presheaf]] $U \mapsto \left\{ \text{finite sums } \sum_{i} s_{i} \cdot \mathcal{f}_{i} ^{-1} |_{U}: U_{i} \supset U_{}, s_{i} \in \mathcal{O}_{X}(U) \right\}.$
> The $\{ U_{i} \}$ form a trivializing cover witnessing that $\mathcal{O}_{X}(D)$ is indeed a [[locally free sheaf|line bundle]], for we have an isomorphism[^1] $\begin{align}
> \mathcal{O}_{X} |_{U_{i}} & \xrightarrow{(-) \cdot f_{i} ^{-1}} \mathcal{O}_{X}(D) |_{U_{i}} \\
> 1 & \mapsto f_{i} ^{-1}
> \end{align}$(inverse is 'multiplication by $f_{i}