Properties of quotient sheaves

Solution 1:

This is a nice example of describing a global section of the sheafification of the (quotient) presheaf by local data that glues. In other words, the sheaf quotient $\mathscr K^\ast / \mathscr O^\ast$ is the sheafification of the presheaf quotient $U\mapsto \mathscr K^\ast(U) / \mathscr O^\ast(U)$, and by the properties of sheaves and definition of sheafification, we know that a global section of $\mathscr K^\ast / \mathscr O^\ast$ is determined by a collection of local sections of the presheaf quotient that glue on overlaps.

Now a local section of the presheaf quotient is just an element $f_i\pmod{ \mathscr O^\ast(U_i)}$, and asking that a collection $\{ f_i \}$ glues is the same as requiring that $f_i = f_j \pmod{\mathscr O^\ast(U_{ij})}$, which is the same as $f_i/f_j \in\mathscr O^\ast(U_{ij})$.