Divisor class group of an A-type surface singularity
= Divisor class group of an A-type surface singularity
{c}
{title2=$\operatorname{Cl}(k[x,y,z]/(xy-z^n))$}
For $n\geq2$, the normal affine surface
$$
X_n=\operatorname{Spec}k[x,y,z]/(xy-z^n)
$$
has $\operatorname{Cl}(X_n)\cong\mathbb Z/n\mathbb Z$. The class of the prime divisor $D=V(x,z)$ generates: localization at $x$ is a unique factorization domain, so the divisor-class localization sequence leaves only $D$, while $\operatorname{div}(x)=nD$ gives its exact order.