Divisor line bundle
= Divisor line bundle
{title2=$\mathcal O_X(D)$}
On an <integral scheme>, for a <Cartier divisor> locally represented by $a_i\in K^*$, its divisor line bundle is the subsheaf of rational functions locally equal to $a_i^{-1}\mathcal O_X$. Unit ratios glue these free rank-one modules. Given a nonzero <rational section of a line bundle> $s=a_ie_i$, the maps $e_i\mapsto a_i^{-1}$ identify that <invertible sheaf> with $\mathcal O_X(D)$. Replacing $s$ by a nonzero rational multiple changes $D$ by a <principal Cartier divisor>.