Fibre degree of the finite locally free map to the projective line

The fibre $C_0:=C{\times }_{{\mathbb{P}}_k^1}\mathrm{Spec}k$ over $0$ is canonically $\mathrm{Spec}(B_0\mathrm{/}t{B}_0)$, and ${\dim }_k\mathrm{\Gamma }(C_0,{\mathcal{O}}_{C_0})=d$. Since $0\in U_0$ and $U_0$ is open, the structure morphism $\mathrm{Spec}k\to {\mathbb{P}}_k^1$ with image $0$ factors through $U_0$, so $C_0\cong {\phi }_f^{-1}(U_0){\times }_{U_0}\mathrm{Spec}k$. By [F10] and [F1] the scheme ${\phi }_f^{-1}(U_0)=\mathrm{Spec}{B}_0$ is affine and $\mathrm{Spec}{B}_0\to U_0=\mathrm{Spec}k[t]$ is affine, so [F7] identifies $C_0$ with $\mathrm{Spec}(B_0{\otimes }_{k[t]}k)$, which is $\mathrm{Spec}(B_0\mathrm{/}t{B}_0)$ by [F8]. Since $B_0$ is a free $k[t]$-module of rank $d$, [F8] gives $B_0\mathrm{/}t{B}_0\cong (k[t]\mathrm{/}(t){)}^d\cong k^d$ as $k$-vector spaces, so the coordinate ring has $k$-dimension $d$. A base change of a finite morphism is finite, so $C_0$ is finite over $\mathrm{Spec}k$ and has finitely many points.

The underlying set of $C_0$ is exactly the set of closed points $x$ of $C$ with ${\mathrm{ord}}_x(f)>0$. By [F7] the points of $C_0$ are the points $x$ of $C$ with ${\phi }_f(x)=0$; since $C_0$ is finite over $k$ by 1.1, its points are closed in $C_0$ and are closed points of the one-dimensional $k$-scheme $C$ (the generic point $\eta$ maps to the generic point of ${\mathbb{P}}_k^1$ because ${\phi }_f$ is nonconstant and $C$ is integral, so $\eta \mathrm{\notin }{C}_0$). A point $x$ maps to $0$ exactly when $x$ lies in ${\phi }_f^{-1}(U_0)$, so that $f$ is regular at $x$, and the image of $f$ in $\kappa (x)$ is zero; for the discrete valuation ring ${\mathcal{O}}_{C,x}$ this is exactly the condition ${\mathrm{ord}}_x(f)>0$.

For every $x\in C_0$ one has ${\mathcal{O}}_{C_0,x}\cong {\mathcal{O}}_{C,x}\mathrm{/}(f)$ and ${\dim }_k{\mathcal{O}}_{C_0,x}={\mathrm{ord}}_x(f)[\kappa (x):k]$. By 1.2 the point $x$ is a closed point with $n:={\mathrm{ord}}_x(f)>0$ and $f=u{\pi }_x^n$ with $u$ a unit of the discrete valuation ring ${\mathcal{O}}_{C,x}$. By [F6] applied to ${\phi }_f$ and the point $0\in {\mathbb{P}}_k^1$, ${\mathcal{O}}_{C_0,x}\cong {\mathcal{O}}_{C,x}\mathrm{/}{\mathfrak{m}}_0{\mathcal{O}}_{C,x}$; since ${\phi }_f^{\mathrm{\#}}(t)=f$, the ideal ${\mathfrak{m}}_0{\mathcal{O}}_{C,x}$ is $(f)$, so ${\mathcal{O}}_{C_0,x}\cong {\mathcal{O}}_{C,x}\mathrm{/}(f)$. By [F5] this is a module of length $n$ over ${\mathcal{O}}_{C,x}$, with a composition series whose $n$ factors are isomorphic to $\kappa (x)$; each factor has $k$-dimension $[\kappa (x):k]$ by [F3], and dimensions add along this filtration of $k$-vector spaces by [F9]. Hence ${\dim }_k{\mathcal{O}}_{C_0,x}=n[\kappa (x):k]$.

The identity ${\sum }_{x\in C_0}{\mathrm{ord}}_x(f)[\kappa (x):k]={\dim }_k\mathrm{\Gamma }(C_0,{\mathcal{O}}_{C_0})=d$ holds. By 1.1 the ring $\mathrm{\Gamma }(C_0,{\mathcal{O}}_{C_0})=B_0\mathrm{/}t{B}_0$ is a finite-dimensional $k$-algebra, hence Artinian, and its maximal ideals are the finitely many points $x\in C_0$ with local rings ${\mathcal{O}}_{C_0,x}$. By [F9] it is the product of those local rings, so its $k$-dimension is the sum of the $k$-dimensions computed in 1.3, namely ${\sum }_{x\in C_0}{\mathrm{ord}}_x(f)[\kappa (x):k]$; by 1.1 this equals $d$. This is the zero-fibre identity.

Pole fibre. The fibre $C_{\mathrm{\infty }}:=C{\times }_{{\mathbb{P}}_k^1}\mathrm{Spec}k$ over the point at infinity is $\mathrm{Spec}(B_1\mathrm{/}u{B}_1)$, ${\dim }_k\mathrm{\Gamma }(C_{\mathrm{\infty }},{\mathcal{O}}_{C_{\mathrm{\infty }}})=d$, its points are exactly the closed points $x$ with ${\mathrm{ord}}_x(f)<0$, and ${\dim }_k{\mathcal{O}}_{C_{\mathrm{\infty }},x}=(-{\mathrm{ord}}_x(f))[\kappa (x):k]$ for such $x$. The point $\mathrm{\infty }$ lies in $U_1=\mathrm{Spec}k[u]$ and has residue field $k$, so the argument of steps 1.1 and 1.2 applies verbatim to the chart $U_1$ and the coordinate $u$, whose pullback is $f^{-1}$: the fibre is $\mathrm{Spec}(B_1{\otimes }_{k[u]}k)=\mathrm{Spec}(B_1\mathrm{/}u{B}_1)$, and $B_1\mathrm{/}u{B}_1\cong k^d$ because $B_1$ is free of rank $d$ over $k[u]$. A point $x$ of $C$ maps to $\mathrm{\infty }$ exactly when $x\in {\phi }_f^{-1}(U_1)$ and $f^{-1}\in {\mathfrak{m}}_x$, i.e. ${\mathrm{ord}}_x(f)<0$; since $f^{-1}=u{\pi }_x^{-n}$ with $-n={\mathrm{ord}}_x(f^{-1})>0$, [F5] gives length $-n$ and step 1.3 gives ${\dim }_k{\mathcal{O}}_{C_{\mathrm{\infty }},x}=(-n)[\kappa (x):k]$. The Artinian product argument of step 1.4 now yields ${\sum }_{x\in C_{\mathrm{\infty }}}(-{\mathrm{ord}}_x(f))[\kappa (x):k]={\dim }_k\mathrm{\Gamma }(C_{\mathrm{\infty }},{\mathcal{O}}_{C_{\mathrm{\infty }}})=d$.

Both displayed identities hold: for every normal proper curve $C\mathrm{/}k$ and every $f\in K^{\times }$ transcendental over $k$, the zero and pole fibres of the finite locally free morphism ${\phi }_f$ have degree $d=[K:k(f)]$, computed respectively as $\sum {\mathrm{ord}}_x(f)[\kappa (x):k]$ and $\sum (-{\mathrm{ord}}_x(f))[\kappa (x):k]$. The Axiom of Choice is used exactly as declared, through the construction input [F1] and the Artinian decomposition input [F9]; no further choice is made.