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For a smooth irreducible algebraic curve, send a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$ to the line bundle associated to a divisor defined locally by ${\mathcal{O}}_X(D{)}_P=t_P^{-n_P}{\mathcal{O}}_{X,P}$. A nonzero rational section of a line bundle gives its inverse construction, and changing that section changes its divisor by a principal divisor. Thus the divisor class group is the Picard group. The quotient of the sheaf of nonzero rational functions on an irreducible variety by the sheaf of units of the structure sheaf has stalk $\mathbb{Z}$ at each closed point, via the discrete valuation. Its global sections are finite sums of points. The corresponding long exact sequence in sheaf cohomology, and the vanishing for the rational-function sheaf, identify both groups with $H^1(X,{\mathcal{O}}_X^{\ast })$.

Put $\mathcal{Q}={\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, the quotient in sheaves of abelian groups, where ${\mathcal{K}}_X^{\ast }$ is the sheaf of nonzero rational functions on an irreducible variety and ${\mathcal{O}}_X^{\ast }$ is the sheaf of units of the structure sheaf. At a closed point, the discrete valuation induces an isomorphismThe kernel consists precisely of the units of the discrete valuation ring, and surjectivity follows by taking powers of a uniformizer.

A global section of $\mathcal{Q}$ is locally represented by nonzero rational functions. Since $X$ has a finite affine cover and is Noetherian, it is quasi-compact, so finitely many such representatives suffice. Each has only finitely many zeros and poles; consequently the valuations of the section define a finite divisor on an algebraic curve. A section with zero valuation at every closed point is zero in every stalk and therefore zero. This gives an injective mapIt is surjective as well. For $D={\sum }_P{n}_{P}P$, take a rational uniformizer $t_P$ at each point of its finite support. On a sufficiently small neighborhood of $P$, the function $t_P^{n_P}$ has the prescribed divisor there: remove the other points in its zero and pole set and the other support points of $D$. On the complement of the support use the rational function $1$. On overlaps these representatives have the same valuations at every point, so their ratios are units everywhere on the overlap, and their classes in $\mathcal{Q}$ agree. The sheaf gluing axiom now supplies a global section. ThusSince $X$ is irreducible, $\Gamma (X,{\mathcal{K}}_X^{\ast })=K^{\ast }$. Under this identification the map to $\Gamma (X,\mathcal{Q})$ is $f\mapsto \mathrm{div}(f)$. Its image is exactly the group of principal divisor elements, and hence

The sheaf of nonzero rational functions on an irreducible variety ${\mathcal{K}}_X^{\ast }$ is a flasque sheaf. Indeed, its sections on any nonempty open subset are $K^{\ast }$: irreducibility makes nonempty open subsets connected and makes any two of them meet, so locally constant rational-function values must agree. Restrictions between such opens are identities; restriction to the empty set is also surjective. The preliminary argument therefore givesApply the long exact sequence in sheaf cohomology to the short exact sequence of sheaves of multiplicative abelian groupsIts relevant part isPart (b) identifies the first map with the principal divisor map $K^{\ast }\to \mathrm{Div}(X)$. Exactness therefore identifies its cokernel with $H^1(X,{\mathcal{O}}_X^{\ast })$. Combining this with part (a) givesThe connecting map sends a divisor to the class of the unit ratios of its local rational representatives; these are precisely the transition data of the associated line bundle, up to the usual choice of inverse transition convention.

The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.