Residues on the projective line and the vanishing of their sum

Assume the Axiom of Choice as inherited from the cited cohomology and residue suppliers (The Axiom of Choice). Let $k$ be a perfect field and let $C={\mathbb{P}}_k^1$ have homogeneous coordinates $x_0,x_1$, affine coordinate $t=x_1\mathrm{/}{x}_0$ on $U_0=\mathrm{Spec}k[t]$, and point at infinity $\mathrm{\infty }=[0:1]$. Every rational differential is $f(t)dt$ with $f\in k(t)$.

(1) Finite points. Let $p=V(g)$ be a closed point of $U_0$ defined by a monic irreducible $g\in k[t]$, and put $u=g(t)$. Since $k$ is perfect, $g$ is separable, $g^{\mathrm{\prime }}(t)$ is a unit at $p$, and $u$ is a local parameter. If the completed local expansion is $f(t)={\sum }_{n\ge N}{a}_n{u}^n$ with $a_n\in \kappa (p)$, then the coefficient-trace definition gives ${\mathrm{res}}_p(f(t)dt)={\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}\text{\hspace{0.17em}⁣}([u^{-1}]\frac{f(t)}{g^{\mathrm{\prime }}(t)})\mathrm{.}$ Here $1\mathrm{/}{g}^{\mathrm{\prime }}(t)$ is expanded as a unit power series in $u$, since $du=g^{\mathrm{\prime }}(t)dt$. For a simple pole $f=c\mathrm{/}u+\text{regular}$ this is ${\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}(c\mathrm{/}{g}^{\mathrm{\prime }}(\bar{t}))$; for higher-order poles the positive powers in the unit expansion can also contribute. When $k$ is algebraically closed and $p=a$ is a rational point, this is the coefficient of $(t-a{)}^{-1}$ in the Laurent expansion of $f$.

(2) The point at infinity. On $U_1=\mathrm{Spec}k[s]$ with $s=x_0\mathrm{/}{x}_1=1\mathrm{/}t$, the local parameter is $s$ and $t=s^{-1},dt=-s^{-2}ds\mathrm{.}$ Thus ${\mathrm{res}}_{\mathrm{\infty }}(fdt)$ is the coefficient of $s^{-1}$ in $-f(1\mathrm{/}s)s^{-2}ds$. In particular, ${\mathrm{res}}_{\mathrm{\infty }}(t^{n}dt)=\{\begin{array}{ll}-1,& n=-1,\\ 0,& n\mathrm{\ne }-1.\end{array}$

(3) Vanishing of the sum. The monomials $t^{n}dt$ have possible poles only at $0$ and $\mathrm{\infty }$; their residues there cancel for $n=-1$ and are both zero otherwise. For every rational differential on ${\mathbb{P}}_k^1$, the sum of its residues at all closed points is zero by the global residue theorem (The global residue theorem on a smooth proper curve over a perfect field). For example, $\frac{dt}{t(t-1)}=-\frac{dt}{t}+\frac{dt}{t-1}$ has residues $-1$ at $0$, $+1$ at $1$, and $0$ at infinity.

(4) The Laurent-tail class. The principal part $t^{-1}dt$ at the origin represents a class $[t^{-1}dt]\in H^1({\mathbb{P}}_k^1,\omega )$, where $\omega ={\mathrm{\Omega }}_{{\mathbb{P}}_k^1\mathrm{/}k}^1$. Its positive residue sum is $+1$, so the class is nonzero. The space $H^1({\mathbb{P}}_k^1,\omega )$ is one-dimensional. Under the fixed Gysin trace of Serre duality, its value is $-1$, the negative of the positive residue sum; in characteristic two these scalars coincide.