Maybe this exercise comes from some textbook, but I do not know.
It said that this ring extension $k[x(x-1),x^2(x-1),z]\subset k[x,z]$ does not have the Going-Down property.
I observe that $k[x(x-1),x^2(x-1)]=\{f(x)\in k[x]|f(0)=f(1)\}$, and $k[x(x-1),x^2(x-1),z]$ is isomorphic to $k[x,y,z]/(y^2-xy-x^3)$. We have a morphism from $\mathrm{Spec}\ k[x,y,z]/(y^2-xy-x^3)$ to $\mathbb{A}^2$.
But I still have not solved the exercise. And I do not know how one found this counterexample. Why did he consider this?
Thanks.