Invertible sheaf of cartier divisor

Let $D$ be a Cartier divisor on a scheme $X$, represented by meromorphic units $f_i\in {\mathcal{K}}_X(U_i{)}^{\times }$ with regular-unit ratios on the overlaps (Cartier divisor). The subsheaf ${\mathcal{O}}_X\subseteq {\mathcal{K}}_X$ is the one of Sheaf total quotient rings. Define an ${\mathcal{O}}_X$-submodule sheaf of ${\mathcal{K}}_X$ by ${\mathcal{O}}_X(D)(V):=\{g\in {\mathcal{K}}_X(V):f_{i}g{\mathrm{\mid }}_{V\cap U_i}\in {\mathcal{O}}_X(V\cap U_i)\text{ for every }i\}\mathrm{.}$ Equivalently, on each chart, ${\mathcal{O}}_X(D){\mathrm{\mid }}_{U_i}=f_i^{-1}{\mathcal{O}}_{U_i}\subseteq {\mathcal{K}}_X{\mathrm{\mid }}_{U_i}\mathrm{.}$ These are meromorphic functions whose possible poles are cancelled by the local equation of $D$.

This construction is well defined. It is closed under addition and regular scalar multiplication. The condition is local, so compatible sections glue in ${\mathcal{K}}_X$ and retain it by the sheaf locality axiom (A sheaf on a topological space). On an overlap the unit $f_i\mathrm{/}{f}_j$ gives $f_i^{-1}{\mathcal{O}}_X=f_j^{-1}{\mathcal{O}}_X$. Replacing equations by unit multiples or restricting to a refinement gives the same subsheaf; any two representations of the same Cartier divisor agree locally in precisely this sense.

On $U_i$, the map ${\mathcal{O}}_{U_i}\to f_i^{-1}{\mathcal{O}}_{U_i}$, $a\mapsto f_i^{-1}a$, has inverse multiplication by $f_i$. It is injective because $f_i$ is a unit in ${\mathcal{K}}_X$ and ${\mathcal{O}}_X\to {\mathcal{K}}_X$ is injective, and it is surjective by the defining formula. Thus the sheaf is locally free of rank one, hence invertible (Invertible sheaves).

The sign convention allows poles along an effective divisor: ${\mathcal{O}}_X(D){\mathrm{\mid }}_{U_i}=f_i^{-1}{\mathcal{O}}_{U_i}$, whereas ${\mathcal{O}}_X(-D){\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$. If $D$ is effective this last subsheaf is exactly its ideal sheaf ${\mathcal{I}}_D$ of Effective cartier divisor. For the zero divisor the equation is $1$ and ${\mathcal{O}}_X(0)={\mathcal{O}}_X$. On the empty scheme the formula gives its unique module sheaf, which satisfies the local rank-one condition vacuously.