Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral.

Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian algebras (k be a field) that satisfy going down theorem, is there any " factorisation result" for such morphims?

think$B$ is of finite type over $A$ (or anyway, I know some examples where it is), but it isn't faithfully flat (since $PB=B$), nor integral (since $A$ is integrally closed and has the same fraction field as $B$). And there's essentially nothing between $A$ and $B$ to make a factorization out of. But $B$ is going down over $A$. $\endgroup$ – Neil Epstein Mar 20 '15 at 15:49