The index of speciality i(D)

Assume the Axiom of Choice for proper-cohomology finiteness, the curve Cartier-to-Weil identification, and the cohomological Riemann-Roch theorem (The Axiom of Choice). AC supplies the Dependent Choice premise of the curve Cartier-to-Weil result through 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), let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) and let ${\mathcal{O}}_C(D)$ be its associated invertible sheaf. The current Finite-dimensionality of the Riemann-Roch space proves that $H^1(C,{\mathcal{O}}_C(D))$ is finite-dimensional under this assumption. The index of speciality of $D$ is the nonnegative integer $i(D):=h^1(C,{\mathcal{O}}_C(D))={\dim }_k{H}^1(C,{\mathcal{O}}_C(D)),$ the first cohomology dimension of the attached invertible sheaf (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 Riemann-Roch dimension l(D)).

The index of speciality depends only on the linear equivalence class of $D$. Indeed, Cartier and Weil divisors agree on a smooth curve identifies the Weil divisors on $C$ with Cartier divisors and preserves principal divisors; Linear equivalence cartier divisors gives the Cartier equivalence relation; and the current Cartier-sheaf and addition-tensor dictionaries (Invertible sheaf of cartier divisor, The sheaf of a Cartier divisor is invertible, Addition of Cartier divisors is tensor product of their sheaves, Rational sections of line bundles are Cartier divisors) identify linearly equivalent divisors with isomorphic invertible sheaves. Their first cohomology groups therefore have the same dimension.

For the zero divisor, $i(0)=h^1(C,{\mathcal{O}}_C)=g(C),$ by Genus via the Euler characteristic. The independent Riemann-Roch for curves: the Euler-characteristic form gives $h^0(C,{\mathcal{O}}_C(D))-h^1(C,{\mathcal{O}}_C(D))={\mathrm{deg}}_k(D)+1-g(C)\mathrm{.}$ By the definitions of $l(D)$ and $i(D)$ this is $l(D)-i(D)={\mathrm{deg}}_k(D)+1-g(C)\mathrm{.}$ Thus $i(D)\ge 0$ gives the Riemann inequality $l(D)\ge {\mathrm{deg}}_k(D)+1-g(C),$ and equality holds exactly when $i(D)=0$.

This definition is deliberately one-sided. No identification of $H^1(C,{\mathcal{O}}_C(D))$ with the sections of a complementary invertible sheaf, and no Serre-duality statement, is made or used here. The Cartier, finite-dimensionality, and Euler-characteristic suppliers named above are present in the working tree as draft or published items as indicated by their frontmatter; their presence alone does not certify a mathematical review.