= Divisor class group and Picard group of a smooth curve
{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 $H^1(X,\mathcal O_X^*)$.
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