Genus via the Euler characteristic

Assume the Axiom of Choice for proper-cohomology finiteness and the global functions theorem (The Axiom of Choice). Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field). The current Finite-dimensionality of the Riemann-Roch space proves under this assumption that $H^1(C,{\mathcal{O}}_C)$ is finite-dimensional. The genus of $C$ is the nonnegative integer $g(C):=h^1(C,{\mathcal{O}}_C)={\dim }_k{H}^1(C,{\mathcal{O}}_C)$ (The Riemann-Roch dimension l(D), Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

The global functions theorem Functions on a proper curve gives $H^0(C,{\mathcal{O}}_C)=k$. Consequently the Euler characteristic $\chi (C,{\mathcal{O}}_C)=h^0(C,{\mathcal{O}}_C)-h^1(C,{\mathcal{O}}_C)$ of Euler characteristic of a coherent sheaf satisfies $\chi (C,{\mathcal{O}}_C)=1-g(C),\text{equivalently}g(C)=1-\chi (C,{\mathcal{O}}_C)\mathrm{.}$ This is the arithmetic genus $p_a(C)=1-\chi ({\mathcal{O}}_C)$ of Genus and arithmetic genus of a curve for a smooth geometrically connected proper curve. The equality of the two definitions here follows from $H^0(C,{\mathcal{O}}_C)=k$ and finite-dimensionality of $H^1$.

For the projective line, the direct published cohomology calculation Top cohomology of projective twists gives $H^1({\mathbb{P}}_k^1,\mathcal{O})=0$ (take projective dimension $1$ and twist $d=0$), so $g({\mathbb{P}}_k^1)=0$. No Serre duality is used in this definition.