The Riemann-Roch dimension l(D)

Assume the Axiom of Choice (The Axiom of Choice). It supplies the Dependent Choice premise of the current curve Cartier-to-Weil result by AC implies DC implies countable choice. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) and let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) with associated invertible sheaf ${\mathcal{O}}_C(D)$. For every such $k$, $C$ and $D$, the dimensions below are finite; define $l(D):={\dim }_{k}L(D)={\dim }_k{H}^0(C,{\mathcal{O}}_C(D))=h^0(D)$ to be the dimension of the Riemann-Roch space, and define $h^i(D):={\dim }_k{H}^i(C,{\mathcal{O}}_C(D)),i\ge 0,$ for the dimensions of the sheaf cohomology groups (Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

For finiteness, ${\mathcal{O}}_C(D)$ is invertible by Invertible sheaf of cartier divisor, The sheaf of a Cartier divisor is invertible and Cartier and Weil divisors agree on a smooth curve. An invertible sheaf is locally free of rank one, hence quasi-coherent and of finite type. The curve is locally Noetherian because it is of finite type over the Noetherian field $k$; therefore a finite-type quasi-coherent sheaf on it is coherent (Invertible sheaves, Locally free sheaves of finite rank, Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Locally finite type and finite type morphisms, 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, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme). The published proper coherent-cohomology theorem then makes every $H^i(C,{\mathcal{O}}_C(D))$ finite-dimensional over $k$ (Finite-dimensional coherent cohomology over a field). Thus $l(D)$ and each $h^i(D)$ are always nonnegative integers in this setting. For the zero divisor, $l(0)={\dim }_k{H}^0(C,{\mathcal{O}}_C)=1,$ because $H^0(C,{\mathcal{O}}_C)$ is canonically $k$ (Functions on a proper curve); more generally $l(D)$ depends only on the divisor $D$ through its associated invertible sheaf.

The current The space L(D) supplies the order-defined space $L(D)$ and identifies it with $H^0(C,{\mathcal{O}}_C(D))$, using the rational-section dictionary Rational sections of line bundles are Cartier divisors. The Cartier-to-Weil route requires Dependent Choice, which the stated Axiom of Choice supplies through AC implies DC implies countable choice. This finiteness route uses the published proper cohomology result directly and does not depend on the later Riemann-Roch-space finite-dimensionality lemma.