The degree of a divisor descends to the Picard group of a normal proper curve

Closed points, prime divisors and local factoriality. Every point $x$ other than the generic point $\eta$ is closed. Indeed, $\overline{\{x\}}$ is a proper irreducible closed subset of $C$; any distinct point $y$ in that closure would give the strict chain $\overline{\{y\}}⊊\overline{\{x\}}⊊C$, contradicting chain dimension one. The first inclusion is strict because points of a scheme with the same closure are equal, as follows on affine spectra from their prime ideals. On an affine neighborhood $\mathrm{Spec}A$ of a closed point $x$, its prime $\mathfrak{p}$ is nonzero and maximal. The chain $(0)⊊\mathfrak{p}$ gives $\dim A_{\mathfrak{p}}\ge 1$ by [F8]. Any longer prime chain would give a longer chain of irreducible closed subsets in this affine open and, by taking closures, in $C$, contradicting [F1]. Thus $\dim {\mathcal{O}}_{C,x}=1$, whereas ${\mathcal{O}}_{C,\eta }=K(C)$ has dimension zero by [F8]. Therefore the prime divisors are precisely the closed points with reduced structure; their local rings are DVRs by [F3]. By [F4] these DVRs, and the field at $\eta$, are UFDs under AC. Hence $C$ is locally factorial, and its Weil divisor group is the finite closed-point divisor group of [F1].

Additivity and principal divisors. The $k$-degree ${\mathrm{deg}}_k:\mathrm{Div}(C)\to \mathbb{Z}$ of [F1] is a group homomorphism, and it annihilates the subgroup $P(C)$ of principal Weil divisors: ${\mathrm{deg}}_k{\mathrm{div}}_W(f)=0$ for every $f\in K(C{)}^{\times }$ by [F6]. Consequently ${\mathrm{deg}}_k$ induces a well-defined group homomorphism $\mathrm{Cl}(C)\to \mathbb{Z}$ on classes, carrying the class $[D]$ of a Weil divisor to ${\mathrm{deg}}_{k}D$.

Every Weil divisor has a Cartier representative, and $\mathrm{Pic}(C)\cong \mathrm{Cl}(C)$. By [F2] and [F3] the curve $C$ is a Noetherian integral scheme, and by step 1.1 it is locally factorial, so [F5] applies: every prime divisor is an effective Cartier divisor, the cycle map $\mathrm{cyc}$ is surjective, and the canonical homomorphism $\phi :\mathrm{Pic}(C)\to \mathrm{Cl}(C)$, $[{\mathcal{O}}_C(D)]\mapsto [\mathrm{cyc}(D)]$, is an isomorphism. In particular a Weil divisor $D$ is the cycle $\mathrm{cyc}(D^{\mathrm{\prime }})$ of some Cartier divisor $D^{\mathrm{\prime }}$, and the sheaf ${\mathcal{O}}_C(D):={\mathcal{O}}_C(D^{\mathrm{\prime }})$ is well defined up to isomorphism: if also $D=\mathrm{cyc}(D^{\mathrm{\prime }\mathrm{\prime }})$, then $\phi ([{\mathcal{O}}_C(D^{\mathrm{\prime }})])=[D]=\phi ([{\mathcal{O}}_C(D^{\mathrm{\prime }\mathrm{\prime }})])$, and injectivity of $\phi$ gives $[{\mathcal{O}}_C(D^{\mathrm{\prime }})]=[{\mathcal{O}}_C(D^{\mathrm{\prime }\mathrm{\prime }})]$.

The degree is well defined on isomorphism classes of line bundles. Let $D,D^{\mathrm{\prime }}$ be Weil divisors on $C$ with ${\mathcal{O}}_C(D)\cong {\mathcal{O}}_C(D^{\mathrm{\prime }})$. Choose Cartier divisors $D_1,D_2$ with $\mathrm{cyc}(D_1)=D$ and $\mathrm{cyc}(D_2)=D^{\mathrm{\prime }}$, as in step 2.1. Then $\phi ([{\mathcal{O}}_C(D_1)])=[D]$ and $\phi ([{\mathcal{O}}_C(D_2)])=[D^{\mathrm{\prime }}]$ by [F5], and $[{\mathcal{O}}_C(D_1)]=[{\mathcal{O}}_C(D)]=[{\mathcal{O}}_C(D^{\mathrm{\prime }})]=[{\mathcal{O}}_C(D_2)]$ in $\mathrm{Pic}(C)$; since $\phi$ is injective, $[D]=[D^{\mathrm{\prime }}]$ in $\mathrm{Cl}(C)$. By [F6] there is $f\in K(C{)}^{\times }$ with $D-D^{\mathrm{\prime }}={\mathrm{div}}_W(f)$, so ${\mathrm{deg}}_{k}D-{\mathrm{deg}}_k{D}^{\mathrm{\prime }}={\mathrm{deg}}_k(D-D^{\mathrm{\prime }})={\mathrm{deg}}_k{\mathrm{div}}_W(f)=0$ by additivity of ${\mathrm{deg}}_k$ in [F1] and vanishing on principal divisors in [F6]. Hence ${\mathrm{deg}}_{k}D$ depends only on the isomorphism class $[{\mathcal{O}}_C(D)]$.

The descended degree is a group homomorphism. Define ${\mathrm{deg}}_k:\mathrm{Pic}(C)\to \mathbb{Z}$ by choosing, for a class $[L]\in \mathrm{Pic}(C)$, the unique class $[D]\in \mathrm{Cl}(C)$ with $\phi ([L])=[D]$ and setting ${\mathrm{deg}}_k[L]:={\mathrm{deg}}_{k}D$; this is independent of all choices by step 3.1 and satisfies ${\mathrm{deg}}_k[{\mathcal{O}}_C(D)]={\mathrm{deg}}_{k}D$ for every Weil divisor $D$ because $\phi ([{\mathcal{O}}_C(D)])=[D]$ by step 2.1. It is additive: if $[L],[L^{\mathrm{\prime }}]\in \mathrm{Pic}(C)$ correspond to $[D],[D^{\mathrm{\prime }}]$, then $[L][L^{\mathrm{\prime }}]=[L\otimes L^{\mathrm{\prime }}]$ corresponds to $[D]+[D^{\mathrm{\prime }}]=[D+D^{\mathrm{\prime }}]$ because $\phi$ is a group homomorphism, so ${\mathrm{deg}}_k([L][L^{\mathrm{\prime }}])={\mathrm{deg}}_k(D+D^{\mathrm{\prime }})={\mathrm{deg}}_{k}D+{\mathrm{deg}}_k{D}^{\mathrm{\prime }}={\mathrm{deg}}_k[L]+{\mathrm{deg}}_k[L^{\mathrm{\prime }}]$ by additivity of ${\mathrm{deg}}_k$ on $\mathrm{Div}(C)$ in [F1]; and ${\mathrm{deg}}_k[{\mathcal{O}}_C]={\mathrm{deg}}_{k}0=0$, so the identity of $\mathrm{Pic}(C)$ is respected. Thus ${\mathrm{deg}}_k{\mathcal{O}}_C(D):={\mathrm{deg}}_{k}D$ is a well-defined group homomorphism $\mathrm{Pic}(C)\to \mathbb{Z}$. ∎