h^1 of a line bundle equals the dimension of the space of dual sections

Given: the Axiom of Choice; a field $k$; a smooth proper geometrically integral curve $C$ over $k$; a canonical divisor $K_C$ on $C$; and a divisor $D$ on $C$.

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

Divisors on $C$ form a free abelian group on the closed points, with support, positive and negative parts and degree ${\mathrm{deg}}_k$; addition and subtraction of divisors are defined componentwise and canonical divisors are divisors of rational differentials (Divisors on a smooth proper curve, Canonical bundle and canonical divisors).

The index of speciality of $D$ is $i(D):=h^1(C,{\mathcal{O}}_C(D))={\dim }_k{H}^1(C,{\mathcal{O}}_C(D))$, computed from the invertible sheaf ${\mathcal{O}}_C(D)$ attached to $D$; it depends only on the linear equivalence class of $D$ and is a nonnegative integer (The index of speciality i(D)).

Over any field $k$, for an invertible ${\mathcal{O}}_C$-module $\mathcal{L}$ on the smooth proper geometrically integral curve $C$ one has $h^1(C,\mathcal{L})=h^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})$, with ${\mathcal{L}}^{-1}={\mathcal{L}}^{\vee }$ the dual (Tensor product of sheaves of modules), and the pairing realizing this equality is perfect and functorial in $\mathcal{L}$ (Serre duality for line bundles on a smooth proper curve, and the residue realization).

For a canonical divisor $K_C$ the canonical bundle satisfies ${\omega }_C\cong {\mathcal{O}}_C(K_C)$ for the invertible sheaf attached to the divisor, by the dictionary between invertible sheaves with a rational section and divisors (Canonical bundle and canonical divisors).

The in-run draft theorem on line bundles and divisors (authored as a draft in this run; its supplier closure remains pending) states: on an integral scheme, every pair $(\mathcal{L},s)$ of an invertible sheaf and a nonzero rational section determines a Cartier divisor ${\mathrm{div}}_C(s)$ with ${\mathcal{O}}_X({\mathrm{div}}_C(s))\cong \mathcal{L}$ carrying its canonical rational section $1$ to $s$; conversely every Cartier divisor arises, and pair isomorphism is the equivalence relation. In particular, applied to $\mathcal{L}={\omega }_C\otimes {\mathcal{O}}_C(-D)$ and a nonzero rational section with divisor $K_C-D$, it gives ${\omega }_C\otimes {\mathcal{O}}_C(-D)\cong {\mathcal{O}}_C(K_C-D)$. This is the promised claim of Rational sections of line bundles are Cartier divisors, treated here as a declared supplier.

The in-run draft definition of the Riemann-Roch space (authored as a draft in this run; its supplier closure remains pending) states: for a divisor $D$ on the smooth proper geometrically integral curve $C$ one sets $L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}$, and under the identification of divisors with invertible sheaves $L(D)=H^0(C,{\mathcal{O}}_C(D))$ as a $k$-subspace of $k(C)$, so that $l(D)={\dim }_{k}L(D)=h^0(C,{\mathcal{O}}_C(D))$. This is the promised claim of The space L(D), treated here as a declared supplier.

Euler-characteristic Riemann-Roch in divisor notation gives $l(D)-i(D)={\mathrm{deg}}_k(D)+1-g$ for every divisor over the arbitrary field (Riemann-Roch as l minus i). Its divisor dictionary suppliers remain the declared in-run draft prerequisites.

Cartier divisor addition and inverse give canonical isomorphisms ${\mathcal{O}}_C(K_C)\otimes {\mathcal{O}}_C(-D)\cong {\mathcal{O}}_C(K_C-D)$ and ${\mathcal{O}}_C(D{)}^{\vee }\cong {\mathcal{O}}_C(-D)$ (Addition of Cartier divisors is tensor product of their sheaves), with the curve divisor dictionary as in [F2, F5, F6].