Cartier divisor local equation equivalence

The quotient sheaf $Q$ is the cokernel of the map of sheaves ${\mathcal{O}}_X^{\times }\to {\mathcal{K}}_X^{\times }$. Indeed the cokernel sheaf is the sheafification of $U\mapsto \mathrm{coker}({\mathcal{O}}_X^{\times }(U)\to {\mathcal{K}}_X^{\times }(U))$, which is exactly the quotient presheaf of [F1].

At every point $x\in X$ one has $Q_x\cong {\mathcal{K}}_{X,x}^{\times }\mathrm{/}{\mathcal{O}}_{X,x}^{\times }$. Let $P(U)={\mathcal{K}}_X^{\times }(U)\mathrm{/}{\mathcal{O}}_X^{\times }(U)$, so $Q=aP$ by [F1]. The map from ${\mathcal{K}}_{X,x}^{\times }\mathrm{/}{\mathcal{O}}_{X,x}^{\times }$ to $Q_x$ sends the class of a germ represented by $f\in {\mathcal{K}}_X^{\times }(U)$ to the germ of the sheafified class of $f$. It is surjective: [F3] identifies $Q_x$ with $P_x$, and by [F4] every element of $P_x$ is represented by a quotient class $[f]\in P(U)$ on a neighbourhood $U$ of $x$. To see injectivity, suppose the class of $f_x$ maps to the identity germ. By [F3] and [F8], after shrinking to a neighbourhood $V$ of $x$, the quotient class $[f{\mathrm{\mid }}_V]$ is the identity class in $P(V)$. By [F5] this means $f{\mathrm{\mid }}_V$ is a section of ${\mathcal{O}}_X^{\times }(V)$, so $f_x$ belongs to ${\mathcal{O}}_{X,x}^{\times }$. Conversely every germ from ${\mathcal{O}}_X^{\times }$ maps to the identity. The subgroup embeds in ${\mathcal{K}}_{X,x}^{\times }$ by [F6], giving the claimed quotient.

The quotient map $q\text{\hspace{0.17em}⁣}:{\mathcal{K}}_X^{\times }\to Q$ has kernel exactly ${\mathcal{O}}_X^{\times }$. For each $x$ the map $q_x$ is the quotient map ${\mathcal{K}}_{X,x}^{\times }\to {\mathcal{K}}_{X,x}^{\times }\mathrm{/}{\mathcal{O}}_{X,x}^{\times }$ by step 1.2, so its kernel is ${\mathcal{O}}_{X,x}^{\times }$. The kernel subsheaf of $q$ therefore has the same stalks as ${\mathcal{O}}_X^{\times }$, and the inclusion of subsheaves is an isomorphism by the stalkwise criterion.

Every section of $Q$ is locally a class of a meromorphic unit. Let $s\in Q(X)$ and $x\in X$. Because $q$ is a cokernel projection it is surjective on stalks, so the germ $s_x$ is the image of some element of ${\mathcal{K}}_{X,x}^{\times }$; that element is represented by a section $f$ of ${\mathcal{K}}_X^{\times }$ over an open neighbourhood $V$ of $x$, and $q(f)$ and $s$ have equal germs at $x$, hence agree on some neighbourhood of $x$ contained in $V$.

Every local-equation datum determines a global section of $Q$. On $U_i\cap U_j$ the ratio $f_i\mathrm{/}{f}_j$ is a unit, so $q(f_i)$ and $q(f_j)$ have equal restriction because their difference is the class of a unit, which vanishes in the quotient. The sections $q(f_i)\in Q(U_i)$ therefore agree on all overlaps and glue by the sheaf axiom to a section $s\in Q(X)$ with $s{\mathrm{\mid }}_{U_i}=q(f_i)$.

Every section of $Q$ is induced by a local-equation datum. Let $s\in Q(X)$ and take the set of all pairs $(V,f)$ with $V\subseteq X$ open, $f\in {\mathcal{K}}_X^{\times }(V)$, and $q(f)=s{\mathrm{\mid }}_V$. By step 3.1, the opens in these pairs cover $X$. For any two such pairs $(V,f)$ and $(W,g)$, the equality of their images with the restrictions of $s$ gives $q(f\mathrm{/}g)=1$ on $V\cap W$. By step 2.1, $f\mathrm{/}g$ is a unit there. Thus this entire indexed family is a local-equation datum; no lift is selected separately for each point.

Two data inducing the same section differ by local units. Let $\{(U_i,f_i)\}$ and $\{(V_j,g_j)\}$ induce the same $s$. The nonempty intersections $W_{ij}=U_i\cap V_j$ form a common refinement. On each such $W_{ij}$, the classes of $f_i$ and $g_j$ agree, so $f_i\mathrm{/}{g}_j$ lies in the kernel of $q$, that is, in ${\mathcal{O}}_X^{\times }(W_{ij})$. Thus $f_i{\mathrm{\mid }}_{W_{ij}}=u_{ij}{g}_j{\mathrm{\mid }}_{W_{ij}}$ for the unit $u_{ij}=f_i\mathrm{/}{g}_j$, and every unit multiple arises this way from another datum.

The sections of $Q$ are exactly the local-equation data modulo refinement and local units.