Divisor class group of the three-dimensional affine quadric cone ($\mathrm{Cl}(k[x,y,z,w]/(xy-zw))\cong \mathbb{Z}$)

For the three-dimensional affine quadric conethe divisor class group is infinite cyclic. A generator is either ruling plane $V(x,z)$ or $V(x,w)$, and the principal divisor of $x$ gives the relation $[V(x,z)]+[V(x,w)]=0$.