Divisors on a smooth proper curve

Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field). A divisor on $C$ is a finite formal $\mathbb{Z}$-linear combination $D={\sum }_{x\in C}{n}_x[x],n_x\in \mathbb{Z},$ of closed points $x$ of $C$, all but finitely many coefficients vanishing.

The finite-formal-sum definition above is unqualified. For the following local-ring and normality context, assume the Axiom of Choice (The Axiom of Choice); it supplies Dependent Choice by AC implies DC implies countable choice. If $x$ is a closed point of $C$, then ${\mathcal{O}}_{C,x}$ is a discrete valuation ring by Local rings at closed points of smooth curves are discrete valuation rings. At the generic point $\eta$, the local ring is the function field $k(C)$, which is a field. These are all the points of this one-dimensional integral curve, and both kinds of local rings are integrally closed domains. Since $C$ is finite type over the field $k$, its affine coordinate rings are Noetherian; the finite-type scheme is quasi-compact, so $C$ is Noetherian (A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes). Thus $C$ is normal (normal noetherian ring). The codimension-one points are exactly the closed points. Hence the finite sums above are the Weil divisors of the fixed normal-scheme convention (Weil divisor normal noetherian scheme).

Under the same Choice assumption, each closed-point local ring is a PID by Every DVR is a PID and a UFD by Every principal ideal domain is a unique factorisation domain; the generic local ring is a field and hence a UFD. Thus $C$ is locally factorial (Locally factorial scheme). This is the local-factorial input for the usual Cartier interpretation of these curve divisors, established by the separate curve-level Cartier/Weil comparison. The divisor group, support, degree, and effectivity conventions used throughout are:

1. the support $\mathrm{Supp}D=\{x:n_x\mathrm{\ne }0\}$ is the finite set of closed points with nonzero coefficient, and the positive and negative parts are $D^{+}={\sum }_{n_x>0}{n}_x[x]$ and $D^{-}={\sum }_{n_x<0}(-n_x)[x]$, so that $D=D^{+}-D^{-}$ (Divisor support positive negative parts);
2. the degree is ${\mathrm{deg}}_k(D)={\sum }_x{n}_x[\kappa (x):k]$, the sum over the finite support of the coefficients weighted by the residue degrees (Degree divisor proper curve); for a closed point of a curve over $k$ the residue field $\kappa (x)$ is a finite extension of $k$;
3. $D$ is effective, written $D\ge 0$, when $n_x\ge 0$ for every $x$; and for two divisors one writes $D\ge D^{\mathrm{\prime }}$ when $D-D^{\mathrm{\prime }}$ is effective.

A divisor is thus an element of the free abelian group on the closed points of $C$, and the divisor of a nonzero rational function, the class group, and the Riemann–Roch space of $D$ are the invariants built from this group later on this page.