Finite-dimensionality of the Riemann-Roch space

Given: a field $k$, a smooth proper geometrically integral curve $C$ over $k$, and a divisor $D$ on $C$.

A curve over $k$ is geometrically integral, separated and of finite type over $k$, and its underlying topological space has chain dimension one; being geometrically integral, $C$ is an integral $k$-scheme, so it is nonempty, reduced and irreducible, and the adjectives smooth and proper mean that the structure morphism $C\to \mathrm{Spec}k$ is smooth and proper (Curves over a field, Integral schemes).

For an integral $k$-scheme of finite type with underlying space of chain dimension one, the underlying space is Noetherian, and every proper closed subset is a finite set of closed points; the chain dimension is the Krull dimension of Chain dimension and the empty-space convention, so a curve has dimension at most one in the sense required for vanishing theorems (Proper closed subsets of a curve are finite, Chain dimension and the empty-space convention).

If $X$ is a Noetherian topological space with $\dim X\le d$ for an integer $d\ge 0$, then $H^q(X,\mathcal{F})=0$ for every sheaf of abelian groups $\mathcal{F}$ on $X$ and every integer $q>d$ (Grothendieck vanishing on a Noetherian space).

If $X$ is a scheme proper over a field $k$ and $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module, then $H^q(X,\mathcal{F})$ is a finite-dimensional $k$-vector space for every $q\ge 0$, and only finitely many of the groups are nonzero: for a finite affine open cover of $X$ with $n$ members, $H^q(X,\mathcal{F})=0$ for every $q\ge n$ (Finite-dimensional coherent cohomology over a field).

An invertible ${\mathcal{O}}_X$-module is locally free of rank one, and a locally free module is quasi-coherent; a locally free module of rank $r$ is of finite type, since on a chart $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r=\widetilde{A^r}$ with $A^r$ a finitely generated module; on a locally Noetherian scheme a quasi-coherent module is coherent if and only if it is of finite type (Invertible sheaves, Locally free sheaves of finite rank, Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Coherent sheaves on a locally Noetherian scheme).

A scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings; a morphism locally of finite type provides, around every point, an affine chart $\mathrm{Spec}B$ with $B$ a finitely generated algebra over the coordinate ring of an affine open of the target; a field is a Noetherian ring, and a finitely generated algebra over a Noetherian ring is Noetherian (Locally finite type and finite type morphisms, Locally Noetherian and Noetherian schemes, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring).

For a coherent module $\mathcal{F}$ on a scheme proper over $k$ whose cohomology is finite-dimensional with only finitely many nonzero groups, the Euler characteristic $\chi (X,\mathcal{F})={\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F})$ is an integer; the dimension ${\dim }_k$ of a finite-dimensional $k$-vector space is a nonnegative integer, and $H^q$ denotes sheaf cohomology of the underlying sheaf of abelian groups (Euler characteristic of a coherent sheaf, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Sheaf cohomology as right derived global sections).

Weil-to-Cartier, invertibility, and the Riemann-Roch space. The divisor $D$ is a Weil divisor on the smooth curve, so Cartier and Weil divisors agree on a smooth curve identifies it with a Cartier divisor. The local-equation construction of Invertible sheaf of cartier divisor defines ${\mathcal{O}}_C(D)$, and The sheaf of a Cartier divisor is invertible proves it is invertible. The actual definition The space L(D) identifies $L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}$ with the image of $H^0(C,{\mathcal{O}}_C(D))$ in $k(C)$, using Rational sections of line bundles are Cartier divisors. These interfaces supply the uses at steps 1.3 and 5.1.

The Axiom of Choice enters through the proper finiteness theorem [F4], the coherence and Noetherian suppliers [F5] and [F6], and the choice premises of the curve Cartier-to-Weil route [F8]. In ZF, AC implies DC by AC implies DC implies countable choice, so the DC premise of [F8] is available from the stated assumption (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain); no further selection is made below.