= Divisor class group of the three-dimensional affine quadric cone
{title2=$\operatorname{Cl}(k[x,y,z,w]/(xy-zw))\cong\mathbb Z$}
Let $A=k[x,y,z,w]/(xy-zw)$ over a <field> $k$. Its <divisor class group> is infinite cyclic. Inverting $x$ gives
$$
A_x\cong k[x,x^{-1},z,w],
$$
a <unique factorization domain>. The <Nagata theorem for divisor class groups> therefore generates $\operatorname{Cl}(A)$ by the two <prime Weil divisors> $D_1=V(x,z)$ and $D_2=V(x,w)$ above $x=0$.
At either generic point, the defining equation makes $x$ a <uniformizer> times a unit, so $\operatorname{div}(x)=D_1+D_2$. To prove that this is the only relation, a <rational function> with divisor supported on $D_1\cup D_2$ restricts to a unit of $A_x$. Every such unit is $c x^r$ with $c\in k^*$ and $r\in\mathbb Z$. Its divisor is $r(D_1+D_2)$, giving
$$
\operatorname{Cl}(A)\cong\mathbb Z^2/\mathbb Z(1,1)\cong\mathbb Z.
$$
Thus either ruling plane generates, with the other representing its negative. Computing the localized unit group rules out an additional torsion relation.
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