Rational sections of line bundles are Cartier divisors

Coefficients of a rational section. Let $\mathcal{A}$ be the set of all pairs $(U,e)$ with $U\subseteq X$ a nonempty open subset and $e\in \mathcal{L}(U)$ a generator of $\mathcal{L}$ over $U$; by [F2] every point of $X$ lies in the first component of a member of $\mathcal{A}$, so these pairs cover $X$, and no choice is used because $\mathcal{A}$ is determined by a formula. By [F1] the sheaf ${\mathcal{K}}_X(\mathcal{L})$ is the constant sheaf with value the one-dimensional $K(X)$-vector space ${\mathcal{L}}_{\eta }$ and the sections over the connected open $U$ are exactly ${\mathcal{L}}_{\eta }$, so for a pair $(U,e)$ the germ $e_{\eta }$ is a nonzero vector and there is a unique $f\in K(X)$ with $s{\mathrm{\mid }}_U=fe$; since $s\mathrm{\ne }0$, this $f$ lies in $K(X{)}^{\times }$. Write $f=f_{(U,e)}$ for this coefficient.

Unit ratios. Let $(U,e),(V,e^{\mathrm{\prime }})$ be pairs of $\mathcal{A}$ with nonempty intersection $W$. Both $e{\mathrm{\mid }}_W$ and $e^{\mathrm{\prime }}{\mathrm{\mid }}_W$ generate $\mathcal{L}{\mathrm{\mid }}_W$, so $e{\mathrm{\mid }}_W=u{e}^{\mathrm{\prime }}{\mathrm{\mid }}_W$ for a unit $u\in {\mathcal{O}}_X(W{)}^{\times }$ by [F2]. Restricting the identities $s{\mathrm{\mid }}_U=f_{(U,e)}e$ and $s{\mathrm{\mid }}_V=f_{(V,e^{\mathrm{\prime }})}{e}^{\mathrm{\prime }}$ to $W$ and substituting gives $f_{(U,e)}u{e}^{\mathrm{\prime }}{\mathrm{\mid }}_W=f_{(V,e^{\mathrm{\prime }})}{e}^{\mathrm{\prime }}{\mathrm{\mid }}_W$ in the one-dimensional $K(X)$-vector space ${\mathcal{L}}_{\eta }$ of [F1]; since the germ $e_{\eta }^{\mathrm{\prime }}$ is nonzero, $f_{(U,e)}u=f_{(V,e^{\mathrm{\prime }})}$ and hence $f_{(U,e)}\mathrm{/}{f}_{(V,e^{\mathrm{\prime }})}=u^{-1}\in {\mathcal{O}}_X(W{)}^{\times }$.

Every Cartier divisor arises. Let $D$ be a Cartier divisor on $X$ with local datum $(U_i,f_i)$ consisting of meromorphic units with unit ratios on overlaps [F3]. Since $X$ is integral each $f_i$ lies in $K(X{)}^{\times }$ and, by [F4], ${\mathcal{O}}_X(D)$ is invertible with generator $f_i^{-1}$ on $U_i$ and canonical rational section $1_D$ satisfying $1_D{\mathrm{\mid }}_{U_i}=f_i\cdot f_i^{-1}$. The coefficient of $1_D$ in the trivialization $f_i^{-1}$ is therefore $f_i$, so ${\mathrm{div}}_C(1_D)$ is represented by the same datum $(U_i,f_i)$ and equals $D$ by [F3]; in particular $1_D\mathrm{\ne }0$ is a rational section and the construction is surjective onto $\mathrm{CaDiv}(X)$.

The Cartier divisor. The coefficients $f_{(U,e)}\in {\mathcal{K}}_X(U{)}^{\times }=K(X{)}^{\times }$ have unit ratios on overlaps by step 2.1, so by [F3] they glue to a global section ${\mathrm{div}}_C(s)\in \mathrm{CaDiv}(X)$, and step 2.1 shows moreover that any two pairs give local equations differing by a unit, so the class is independent of all choices.

The local isomorphisms. Fix a pair $(U,e)\in \mathcal{A}$ and put $f=f_{(U,e)}$. By [F4] the sheaf ${\mathcal{O}}_X(D)$, $D={\mathrm{div}}_C(s)$, is freely generated by $f^{-1}$ over $U$, so there is a unique ${\mathcal{O}}_U$-linear map ${\phi }_{(U,e)}:{\mathcal{O}}_X(D){\mathrm{\mid }}_U\to \mathcal{L}{\mathrm{\mid }}_U$ with ${\phi }_{(U,e)}(f^{-1})=e$; because it carries a generator to a generator, it is an isomorphism. Let $(V,e^{\mathrm{\prime }})$ be a second pair with coefficient $h=f_{(V,e^{\mathrm{\prime }})}$ and put $W=U\cap V$; by step 2.1 we have $h=fu$ and $e=u{e}^{\mathrm{\prime }}$ there for a unit $u$, so $f^{-1}=u{h}^{-1}$ and ${\phi }_{(U,e)}(h^{-1})={\phi }_{(U,e)}(u^{-1}{f}^{-1})=u^{-1}e=e^{\mathrm{\prime }}={\phi }_{(V,e^{\mathrm{\prime }})}(h^{-1})$; the maps agree on the generator $h^{-1}$ of ${\mathcal{O}}_X(D){\mathrm{\mid }}_W$, hence agree on all sections.

Gluing. By step 3.2 the isomorphisms ${\phi }_{(U,e)}$ agree on all intersections, so by [F5] they glue to an ${\mathcal{O}}_X$-linear morphism $\phi :{\mathcal{O}}_X(D)\to \mathcal{L}$, which is an isomorphism because it restricts to an isomorphism on each member of the cover.

Global sections and effectivity. The inclusion ${\mathcal{O}}_X\hookrightarrow {\mathcal{K}}_X$ induces an inclusion $\mathcal{L}\hookrightarrow {\mathcal{K}}_X(\mathcal{L})$, as seen in any frame. Thus $s$ belongs to $\mathrm{\Gamma }(X,\mathcal{L})$ exactly when every frame coefficient $f_{(U,e)}$ belongs to ${\mathcal{O}}_X(U)$: these local sections then glue by [F5]. Each such coefficient has a nonzero germ at every point of $U$, since a zero germ would make it vanish on a nonempty open containing $\eta$, contrary to $f_{(U,e)}\mathrm{\ne }0$ in $K(X)$. The local rings are domains, so multiplication by these germs is injective. By [F6], this is equivalent to $D={\mathrm{div}}_C(s)$ being effective. Conversely, any effective local equation differs from a frame coefficient by a regular unit, so all frame coefficients lie in ${\mathcal{O}}_X$ and $s$ is a global section of $\mathcal{L}$.

The canonical section maps to $s$. The constant meromorphic function $1$ is the rational section $1_D$ of ${\mathcal{O}}_X(D)$ with $1_D{\mathrm{\mid }}_U=f\cdot f^{-1}$ for every pair $(U,e)$ by [F4]. Under the induced map ${\mathcal{K}}_X(\phi )$ its restriction to $U$ maps to ${\mathcal{K}}_X({\phi }_{(U,e)})(f{f}^{-1})=f{\phi }_{(U,e)}(f^{-1})=fe=s{\mathrm{\mid }}_U$ by step 3.2, so ${\mathcal{K}}_X(\phi )(1_D)=s$ by the uniqueness in [F5]. Hence $(\mathcal{L},s)\cong ({\mathcal{O}}_X(D),1_D)$, which proves 2.

Pair isomorphism and equality of divisors. Let $s\in \mathrm{\Gamma }(X,{\mathcal{K}}_X(\mathcal{L}))$ and $s^{\mathrm{\prime }}\in \mathrm{\Gamma }(X,{\mathcal{K}}_X({\mathcal{L}}^{\mathrm{\prime }}))$ be rational sections with ${\mathrm{div}}_C(s)=D$ and ${\mathrm{div}}_C(s^{\mathrm{\prime }})=D^{\mathrm{\prime }}$. If $D=D^{\mathrm{\prime }}$, steps 3.2, 4.1 and 5.1 produce isomorphisms $\phi :{\mathcal{O}}_X(D)\to \mathcal{L}$ with $\phi (1_D)=s$ and ${\phi }^{\mathrm{\prime }}:{\mathcal{O}}_X(D)\to {\mathcal{L}}^{\mathrm{\prime }}$ with ${\phi }^{\mathrm{\prime }}(1_D)=s^{\mathrm{\prime }}$, so $\psi ={\phi }^{\mathrm{\prime }}\circ {\phi }^{-1}:\mathcal{L}\to {\mathcal{L}}^{\mathrm{\prime }}$ is an isomorphism with $\psi (s)=s^{\mathrm{\prime }}$. Conversely, if $\psi :\mathcal{L}\to {\mathcal{L}}^{\mathrm{\prime }}$ is an isomorphism with $\psi (s)=s^{\mathrm{\prime }}$ and $(U,e)$ is a pair for $\mathcal{L}$ with coefficient $f$, then $(U,\psi (e))$ is a pair for ${\mathcal{L}}^{\mathrm{\prime }}$ and $\psi (s){\mathrm{\mid }}_U=\psi (fe)=f\psi (e)$ shows that its coefficient is again $f$; hence the two families of local equations define the same Cartier divisor, that is, ${\mathrm{div}}_C(s)={\mathrm{div}}_C(s^{\mathrm{\prime }})$. Therefore ${\mathrm{div}}_C$ is a bijection from isomorphism classes of pairs to Cartier divisors.

Conclusion. Assertion 1 is step 3.1, assertion 2 is steps 3.2, 4.1 and 5.1, assertion 3 is step 2.2, and assertion 4 is step 6.1; this proves the theorem.