The Cartier-to-Weil map respects addition and principal divisors

Given: a normal Noetherian integral scheme $X$, with its groups $\mathrm{CaDiv}(X)$, $\mathrm{Div}(X)$ and $\mathrm{Cl}(X)=\mathrm{Div}(X)\mathrm{/}P(X)$, and the assignment $\mathrm{cyc}$.

Assume DC. For a normal Noetherian scheme $X$ and a Cartier divisor $D$ represented by local equations $f_i\in {\mathcal{K}}_X(U_i{)}^{\times }$ on an open cover, and for every prime divisor $Z$ with generic point $\xi$ and every index $i$ with $\xi \in U_i$, the value $v_{\xi }(f_{i,\xi })$ of the normalized valuation of the discrete valuation ring ${\mathcal{O}}_{X,\xi }$ is independent of $i$ and of the local-equation datum; the sum $\mathrm{cyc}(D)={\sum }_Z{v}_{\xi }(f_{i,\xi })[Z]$ is a well-defined Weil divisor on $X$; and if $X$ is integral then $\mathrm{cyc}({\mathrm{div}}_C(f))={\mathrm{div}}_W(f)$ for every $f\in K(X{)}^{\times }$ (Cartier divisors on a normal Noetherian scheme give Weil divisors).

For a prime divisor $Z$ with generic point $\xi$ the order of vanishing is ${\mathrm{ord}}_Z(f)=v_{\xi }(f_{\xi })$, where $v_{\xi }$ is the normalized discrete valuation of ${\mathcal{O}}_{X,\xi }$; $v_{\xi }$ is a group homomorphism $K^{\times }\to \mathbb{Z}$, so ${\mathrm{ord}}_Z(fg)={\mathrm{ord}}_Z(f)+{\mathrm{ord}}_Z(g)$, ${\mathrm{ord}}_Z(1)=0$ and ${\mathrm{ord}}_Z(f^{-1})=-{\mathrm{ord}}_Z(f)$ (Order codimension one rational function).

Cartier divisors on $X$ form an abelian group $\mathrm{CaDiv}(X)$: a sum $D+E$ is represented on a common open cover by the products $f_i{g}_i$ of local equations of $D$ and $E$, passing to a refinement or replacing equations by unit multiples does not change the divisor, and the zero element is the class of the constant equation $1$ (Cartier divisor).

For a global meromorphic unit $f\in K(X{)}^{\times }$ the principal Cartier divisor ${\mathrm{div}}_C(f)=q_X(f)$ is represented by the single global equation $f$ on the open set $X$, and ${\mathrm{div}}_C(fg)={\mathrm{div}}_C(f)+{\mathrm{div}}_C(g)$; the principal Cartier divisors form a subgroup of $\mathrm{CaDiv}(X)$ (Principal cartier divisor).

Assume DC. The principal Weil divisor ${\mathrm{div}}_W(f)={\sum }_Z{\mathrm{ord}}_Z(f)[Z]$ defines a group homomorphism ${\mathrm{div}}_W:K(X{)}^{\times }\to \mathrm{Div}(X)$ whose image is the subgroup $P(X)$ of principal Weil divisors; the class group is $\mathrm{Cl}(X)=\mathrm{Div}(X)\mathrm{/}P(X)$, and two Weil divisors have the same class exactly when their difference is ${\mathrm{div}}_W(f)$ for some $f\in K(X{)}^{\times }$ (Principal weil divisor and class group).

A Weil divisor on a Noetherian normal scheme is a formal sum ${\sum }_Z{n}_Z[Z]$ over the prime divisors with locally finite support, and addition is coefficientwise; in particular two Weil divisors are equal if and only if their coefficients at every prime divisor agree (Weil divisor normal noetherian scheme).

For a normal subgroup $N⊴G$ the quotient group $G\mathrm{/}N$ has product $(gN)(hN)=ghN$, and the quotient map $G\to G\mathrm{/}N$ is a homomorphism (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$).

If $N⊴G$ and $f:G\to H$ is a homomorphism with $N\subseteq \ker f$, then $f$ factors uniquely as $f=\bar{f}\circ \pi$ with $\bar{f}:G\mathrm{/}N\to H$ a homomorphism, $\bar{f}(gN)=f(g)$ (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

On an integral scheme $X$ the rule $D\mapsto [{\mathcal{O}}_X(D)]$ is a group homomorphism $\mathrm{CaDiv}(X)\to \mathrm{Pic}(X)$ with kernel the principal Cartier divisors, and it is surjective; hence the induced map $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\to \mathrm{Pic}(X)$ is an isomorphism of abelian groups (On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group).

The Picard group $\mathrm{Pic}(X)$ is the group of isomorphism classes of invertible ${\mathcal{O}}_X$-modules under tensor product, with identity $[{\mathcal{O}}_X]$ (Picard group of a scheme).

The Axiom of Dependent Choice (DC) is the statement about entire relations and sequences recorded in The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain; the statements [F1] and [F5] are available under it, and no other choice principle is used here.

A subgroup of an abelian group is normal, and a composite of group homomorphisms is a group homomorphism (Normal subgroup: invariance under conjugation, Monoid homomorphism and group homomorphism).