The space L(D)

Assume the Axiom of Choice for the supplied local-order and Cartier/Weil routes below (The Axiom of Choice). It supplies Dependent Choice by AC implies DC implies countable choice. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ with function field $k(C)$ (Curves over a field), and let $D={\sum }_x{n}_x[x]$ be a divisor on $C$, a finite formal $\mathbb{Z}$-linear combination of closed points (Divisors on a smooth proper curve). Every closed point $x$ of $C$ has a well-defined order ${\mathrm{ord}}_x\text{\hspace{0.17em}⁣}:k(C{)}^{\times }\to \mathbb{Z}$, the discrete valuation of the local ring ${\mathcal{O}}_{C,x}$, a discrete valuation ring (Local rings at closed points of smooth curves are discrete valuation rings, Order codimension one rational function), and the divisor of a nonzero rational function $f$ is $\mathrm{div}(f)={\sum }_x{\mathrm{ord}}_x(f)[x]$ (Order codimension one rational function). This sum has finite support: the curve-level Cartier/Weil route identifies it with the cycle of the principal Cartier divisor, whose support is locally finite and hence finite on the quasi-compact curve (Cartier and Weil divisors agree on a smooth curve).

The Riemann-Roch space of the divisor $D$, also called the space $L(D)$, is the subset $L(D)=\{f\in k(C{)}^{\times }:\mathrm{div}(f)+D\ge 0\}\cup \{0\}\subseteq k(C),$ where the inequality is read coefficientwise: ${\mathrm{ord}}_x(f)+n_x\ge 0$ for every closed point $x$. If $n_x<0$, this condition requires a zero of order at least $-n_x$ at $x$; if $n_x>0$, it permits a pole of order at most $n_x$. This is a $k$-subspace of $k(C)$ (Vector space over a field): it contains $0$ by definition; it is closed under addition, because ${\mathrm{ord}}_x(f+g)\ge \min \{{\mathrm{ord}}_x(f),{\mathrm{ord}}_x(g)\}\ge -n_x$ for $f,g\in L(D)$ by the valuation inequality in the discrete valuation ring ${\mathcal{O}}_{C,x}$ (Local rings at closed points of smooth curves are discrete valuation rings), with ${\mathrm{ord}}_x(0)$ read as $+\mathrm{\infty }$ so that a summand $0$ causes no constraint; and it is closed under scalar multiplication, because ${\mathrm{ord}}_x(cf)={\mathrm{ord}}_x(f)$ for $c\in k^{\times }$ and $0\cdot f=0$. In particular $L(D)$ is determined by $D$ and consists of the rational functions that are regular where $n_x=0$, may have poles of order at most $n_x$ where $n_x>0$, and must vanish to order at least $-n_x$ where $n_x<0$.

Equivalence with the space of global sections (promised clause). By Cartier and Weil divisors agree on a smooth curve the divisor $D$ is Cartier. The associated sheaf is the subsheaf ${\mathcal{O}}_C(D)\subseteq {\mathcal{K}}_C$ described in Invertible sheaf of cartier divisor. Write $H^0(C,{\mathcal{O}}_C(D))=\mathrm{\Gamma }(C,{\mathcal{O}}_C(D))$ for its global sections as in Sheaf cohomology as right derived global sections. On a local-equation cover $(U_i,t_i)$ for $D$ it satisfies ${\mathcal{O}}_C(D){\mathrm{\mid }}_{U_i}=t_i^{-1}{\mathcal{O}}_{U_i}\subseteq {\mathcal{K}}_C{\mathrm{\mid }}_{U_i}\mathrm{.}$ Since $C$ is integral, ${\mathcal{K}}_C$ is the constant sheaf with value $k(C)$. Thus any global section of ${\mathcal{O}}_C(D)$ has a single generic value $f\in k(C)$, and all its local restrictions are that same rational function. The zero section corresponds to $f=0$. For $f\mathrm{\ne }0$, the Cartier-to-Weil compatibility in Cartier and Weil divisors agree on a smooth curve says that the order of $t_i$ at a closed point $x\in U_i$ is the coefficient $n_x$ of $D$. Therefore $f$ is a global section exactly when each $t_{i}f$ is regular, which at every closed point is the condition ${\mathrm{ord}}_x(t_{i}f)=n_x+{\mathrm{ord}}_x(f)\ge 0.$ Conversely, if these inequalities hold, then $t_{i}f$ lies in the local ring at every point of $U_i$; at the generic point it is already an element of the function field. The resulting local regular representatives agree as the same element of $k(C)$ and glue on $U_i$. Consequently the canonical inclusion $H^0(C,{\mathcal{O}}_C(D))\hookrightarrow k(C)$ has image exactly $L(D)$, and the inclusion and its inverse are $k$-linear. The local sheaf formula and the rational-section divisor dictionary are also supplied by the current bodies of Invertible sheaf of cartier divisor and Rational sections of line bundles are Cartier divisors.