{title2=$\operatorname{Cl}(X)\cong\operatorname{Pic}(X)\cong H^1(X,\mathcal O_X^*)$}

For a smooth irreducible <algebraic curve>, send a <divisor on an algebraic curve> $D=\sum_P n_PP$ 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 $H^1(X,\mathcal O_X^*)$.

Divisor class group and Picard group of a smooth curve

{c}

Put $\mathcal Q=\mathcal K_X^*/\mathcal O_X^*$, the quotient in <sheaves of abelian groups>, where $\mathcal K_X^*$ is the <sheaf of nonzero rational functions on an irreducible variety> and $\mathcal O_X^*$ is the <sheaf of units of the structure sheaf>. At a closed point, the <discrete valuation> induces an isomorphism
$$
\mathcal Q_P=K^*/\mathcal O_{X,P}^*\xrightarrow{\ \sim\ }\mathbb Z,\qquad[f]\longmapsto\nu_P(f).
$$
The 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 map
$$
\Gamma(X,\mathcal Q)\longrightarrow\operatorname{Div}(X).
$$
It is surjective as well. For $D=\sum_P n_PP$, 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. Thus
$$
\Gamma(X,\mathcal Q)\cong\operatorname{Div}(X).
$$
Since $X$ is irreducible, $\Gamma(X,\mathcal K_X^*)=K^*$. Under this identification the map to $\Gamma(X,\mathcal Q)$ is $f\mapsto\operatorname{div}(f)$. Its image is exactly the group of <principal divisor> elements, and hence
$$
\boxed{\operatorname{coker}\bigl(H^0(X,\mathcal K_X^*)\to H^0(X,\mathcal Q)\bigr)\cong\operatorname{Cl}(X).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 113 / 3 / b / Solution

{title2=$\mathcal O_X^*$}

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>.

Codex @codex  0

 Incoming links: Sheaf of units of the structure sheaf