Fibres, pullbacks and degrees of divisors under a finite morphism of curves

By [F2] the morphism $f$ is surjective and finite and $n=[k(C):k(D)]\ge 1$; the local rings of closed points on $C$ and $D$ are DVRs, the generic local rings are their function fields, and every point is generic or closed by [F3]. Divisors and their degrees are as in [F1].

(Flatness at a closed point.) Let $p\in C$ be closed and put $q=f(p)$. Since $f$ is finite [F2], it is a closed map by [F9]; hence $\{q\}=f(\{p\})$ is closed. The map ${\mathcal{O}}_{D,q}\to {\mathcal{O}}_{C,p}$ is the restriction of the injective function-field map $k(D)\hookrightarrow k(C)$ from [F2], and is therefore injective. As ${\mathcal{O}}_{C,p}$ is a domain, it is torsion-free as an ${\mathcal{O}}_{D,q}$-module. The source local ring is a DVR and hence a PID by [F3], so [F8] makes this module flat. No finite-generation assertion about the individual stalk is needed.

Dominance sends the generic point ${\eta }_C$ to ${\eta }_D$, so the stalk map is the field extension $k(D)\hookrightarrow k(C)$ by [F2]; its target is a flat $k(D)$-module. By [F3], every point of $C$ is either generic or closed, so this and step 1.2 cover every point. The stalkwise definition [F11] of flatness therefore makes $f$ flat.

Because $f$ is flat, every Cartier divisor on $D$ pulls back to a Cartier divisor on $C$, and pullback is additive [F4]; since Cartier and Weil divisors agree on the smooth curves $C,D$ [F6], the pullback $f^{\ast }E$ is defined as a divisor on $C$ and satisfies $f^{\ast }(E+E^{\mathrm{\prime }})=f^{\ast }E+f^{\ast }{E}^{\mathrm{\prime }}$ for divisors $E,E^{\mathrm{\prime }}$ on $D$.

(Pullback of a point.) In the Cartier representative of $[q]$, choose a neighbourhood $U$ of $q$ with a local equation $t$ whose germ is a uniformizer of ${\mathcal{O}}_{D,q}$, shrinking $U$ so its zero locus there is $q$; on $D\setminus \{q\}$ the local equation is $1$ [F4, F6]. These charts give $[q]$. The pullback uses equations $f^{\mathrm{\#}}t$ over $f^{-1}U$ and $1$ over $f^{-1}(D\setminus \{q\})$ [F4]. Since $f({\eta }_C)={\eta }_D\mathrm{\ne }q$, the fibre is a proper closed subset of $C$; [F3] says each of its points is closed. At each such $p$, the order of $f^{\mathrm{\#}}t$ is $e_p$ by [F4]. At a closed point outside the fibre the pulled-back local equation is $1$, of order $0$. The Cartier-to-Weil cycle reads these local orders as coefficients [F4, F10], and the fibre is finite by [F2], so $f^{\ast }[q]={\sum }_{p\in f^{-1}(q)}{e}_p[p]$.

(Degree of the pullback of a point.) Using [F1] to evaluate the degree of the divisor displayed in step 4.1 and the tower law of [F5] for the finite extensions $\kappa (p)\mathrm{/}\kappa (q)\mathrm{/}k$, one has ${\mathrm{deg}}_k(f^{\ast }[q])={\sum }_{p\in f^{-1}(q)}{e}_p[\kappa (p):k]={\sum }_{p\in f^{-1}(q)}{e}_p[\kappa (p):\kappa (q)][\kappa (q):k]=n[\kappa (q):k]=n{\mathrm{deg}}_k([q])$, the third equality being the fibre-degree sum of [F5] and the last [F1].

(Arbitrary divisors.) Write $E={\sum }_{i=1}^m{m}_i[q_i]$ with closed points $q_i$ and nonzero integers $m_i$, a finite sum by [F1]; additivity of pullback [F4] together with step 3.1 gives $f^{\ast }E={\sum }_i{m}_i{f}^{\ast }[q_i]$, and additivity of ${\mathrm{deg}}_k$ [F1] together with step 5.1 gives ${\mathrm{deg}}_k(f^{\ast }E)={\sum }_i{m}_i{\mathrm{deg}}_k(f^{\ast }[q_i])=n{\sum }_i{m}_i{\mathrm{deg}}_k([q_i])=n{\mathrm{deg}}_k(E)$.

(Invertible sheaves and their degrees.) Define $\mathrm{deg}(M)={\mathrm{deg}}_k(E)$ when $M\cong {\mathcal{O}}_D(E)$. This is well-defined: [F6] says that two such divisors differ by a principal divisor, whose degree is zero by [F12]; the same reasoning applies on $C$. Let $M$ be an invertible ${\mathcal{O}}_D$-module; by [F6] there is a divisor $E$ on $D$ with $M\cong {\mathcal{O}}_D(E)$ and $\mathrm{deg}(M)={\mathrm{deg}}_k(E)$, and $f^{\ast }M\cong f^{\ast }{\mathcal{O}}_D(E)\cong {\mathcal{O}}_C(f^{\ast }E)$ by [F6], so $\mathrm{deg}(f^{\ast }M)={\mathrm{deg}}_k(f^{\ast }E)=n{\mathrm{deg}}_k(E)=n\mathrm{deg}(M)$ by step 6.1.

The Axiom of Choice [F7] enters through the finite-morphism, curve-point, DVR and divisorial suppliers used above; [F10] supplies Dependent Choice where the Cartier-to-Weil cycle is used. Together steps 1.2, 2.1, 3.1, 5.1 and 6.1 prove all the claims.