Riemann-Roch for curves: the Euler-characteristic form

Given: a field $k$, a smooth proper geometrically integral curve $C$ over $k$ with genus $g=g(C)$, and a divisor $D$ on $C$.

The curve $C$ is proper, separated and of finite type over the field $k$, geometrically integral and of chain dimension one, so the Euler characteristic of coherent sheaves on $C$ is defined; a divisor on $C$ is a finite formal integral combination of closed points and ${\mathrm{deg}}_k:\mathrm{Div}(C)\to \mathbb{Z}$ is a group homomorphism (Curves over a field, Divisors on a smooth proper curve, Degree divisor proper curve, Euler characteristic of a coherent sheaf).

The genus: $g=g(C)=h^1(C,{\mathcal{O}}_C)={\dim }_k{H}^1(C,{\mathcal{O}}_C)$ and, since $H^0(C,{\mathcal{O}}_C)$ is canonically $k$ with Euler characteristic $\chi (C,{\mathcal{O}}_C)=h^0(C,{\mathcal{O}}_C)-h^1(C,{\mathcal{O}}_C)$, one has $\chi (C,{\mathcal{O}}_C)=1-g$, equivalently $g=1-\chi (C,{\mathcal{O}}_C)$ (Genus via the Euler characteristic, 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 degree shift: for every divisor $D^{\mathrm{\prime }}$ on $C$, $\chi (C,{\mathcal{O}}_C(D^{\mathrm{\prime }}))-\chi (C,{\mathcal{O}}_C)={\mathrm{deg}}_k(D^{\mathrm{\prime }})$, and equivalently $h^0(D^{\mathrm{\prime }})-h^1(D^{\mathrm{\prime }})={\mathrm{deg}}_k(D^{\mathrm{\prime }})+h^0(0)-h^1(0)$; no Serre duality is used (Riemann-Roch in Euler-characteristic form: the degree shift).

Finiteness and the dimension form: for every divisor $D^{\mathrm{\prime }}$ the sheaf ${\mathcal{O}}_C(D^{\mathrm{\prime }})$ is coherent, $H^q(C,{\mathcal{O}}_C(D^{\mathrm{\prime }}))$ is finite-dimensional over $k$ for every $q\ge 0$ and vanishes for $q\ge 2$, and $\chi (C,{\mathcal{O}}_C(D^{\mathrm{\prime }}))=h^0(D^{\mathrm{\prime }})-h^1(D^{\mathrm{\prime }})$ (Finite-dimensionality of the Riemann-Roch space, The Riemann-Roch dimension l(D)).

The current Cartier-divisor interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve supply the attachment of ${\mathcal{O}}_C(D)$, the global-section identification, and ${\mathcal{O}}_C(0)\cong {\mathcal{O}}_C$ used in [F3].

The Axiom of Choice is used exactly through the degree shift [F3], the genus definition [F2] and the finiteness supplier [F4], which inherit it from the proper-cohomology suppliers; no further selection is made below (The Axiom of Choice).