= Divisor class group of an A-type surface singularity
{c}
{title2=$\operatorname{Cl}(k[x,y,z]/(xy-z^n))$}
For $n\ge2$, the normal affine <surface>
$$
X_n=\operatorname{Spec}A_n,
\qquad A_n=k[x,y,z]/(xy-z^n),
$$
has <divisor class group> $\mathbb Z/n\mathbb Z$. Inverting $x$ gives $A_n[x^{-1}]\cong k[x,x^{-1},z]$, a <unique factorization domain>. The only <prime Weil divisor> over $x=0$ is $D=V(x,z)$, so the <Nagata theorem for divisor class groups> makes $[D]$ a generator.
At the <generic point> of $D$, $y$ is a unit and $x=z^n/y$. The local <discrete valuation ring> has <uniformizer> $z$, giving $\operatorname{div}(x)=nD$. This proves that the order divides $n$; exactness requires a further step. If $mD=\operatorname{div}(q)$, then $q$ is a unit on $X_n\setminus D$. The units of $k[x,x^{-1},z]$ are precisely $c x^r$, so $mD=rnD$. Thus every relation is a multiple of $nD$, proving exact order $n$.
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