The full Riemann-Roch theorem on the projective line, in every degree

Given: the Axiom of Choice and its consequence Dependent Choice; a field $k$, the projective line $C={\mathbb{P}}_k^1$ with point at infinity $\mathrm{\infty }$, and a divisor $D$ of degree $d={\mathrm{deg}}_{k}D$.

Every divisor on ${\mathbb{P}}_k^1$ is linearly equivalent to $({\mathrm{deg}}_{k}D)[\mathrm{\infty }]$, and the associated invertible sheaf of $[\mathrm{\infty }]$ is $\mathcal{O}(1)$; the divisor and invertible-sheaf dictionaries agree on ${\mathbb{P}}_k^1$. (Divisors on the projective line are classified by degree, Cartier and Weil divisors agree on a smooth curve, Invertible sheaf of cartier divisor)

$\mathrm{Pic}({\mathbb{P}}_k^1)\cong \mathbb{Z}$, the class of $\mathcal{O}(1)$ corresponding to $1$; hence $\mathcal{O}(D)\cong \mathcal{O}(d)$ for $D$ of degree $d$, and $\mathrm{\ell }(D)=h^0(C,\mathcal{O}(D))$ is the dimension of the Riemann-Roch space of $D$. (The Picard group of the projective line, The Riemann-Roch dimension l(D))

On ${\mathbb{P}}_k^1$, $H^0({\mathbb{P}}_k^1,\mathcal{O}(m))$ is the degree-$m$ part of $k[x_0,x_1]$, of dimension $m+1$ for $m\ge 0$, and is $0$ for $m<0$; the twisting sheaves $\mathcal{O}(m)$ are the ones attached to the standard charts. (Cohomology of O(d) on projective space, Twisting sheaf on Proj)

For a smooth proper geometrically integral curve of genus $g$, the canonical divisor has degree $2g-2$; on ${\mathbb{P}}_k^1$ the genus is $0$, so ${\mathrm{deg}}_k{K}_C=-2$, and $\mathcal{O}(K_C)={\omega }_C$ with $K_C\sim -2[\mathrm{\infty }]$ because every degree-$(-2)$ divisor is linearly equivalent to $-2[\mathrm{\infty }]$. (The canonical divisor has degree 2g - 2, Canonical bundle and canonical divisors, Degree divisor proper curve, [F1])

The full Riemann-Roch theorem states $\mathrm{\ell }(D)-\mathrm{\ell }(K_C-D)={\mathrm{deg}}_k(D)+1-g$ for a divisor $D$ on a smooth proper geometrically integral curve of genus $g$ over an arbitrary field, and the index of speciality is $i(D)=\mathrm{\ell }(K_C-D)$, with $D$ nonspecial when $i(D)=0$. (The full Riemann-Roch theorem for divisors on a smooth proper curve, The index of speciality i(D), Special and nonspecial divisors)

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil divisor dictionary used in [F1] and [F4]. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)