Properties of quotient sheaves

Question

I am reading Hartshorne's Algebraic Geometry, II.6 about Cartier Divisor. It is defined to be the global section of the sheaf $K^*/O^*$. Then it said: " thinking of the properties of the quotient sheaves, we see that a Cartier divisor can be described by an open cover $U_i$ of $X$, and for each $i$ an element $f_i \in \Gamma(U_i, K^*)$ such that for $i,j$, $f_i/f_j \in \Gamma(U_i \cap U_j, O^*)$. I couldn't make sense of this part. Can someone explain more for me please? So, we need to take the presheaf $U \rightarrow K^*(U)/O^*(U)$, and make it a sheaf? I am always uncertain about how to deal with sheafification.

Answer 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 $f_i/f_j \in\mathscr O^\ast(U_{ij})$.