# going down theorem

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?

• What sort of factorization do you want? – Neil Epstein Mar 20 '15 at 15:28
• for example faithfully flat followed by integral. – prochet Mar 20 '15 at 15:31
• Then no. Let $A$ be a Dedekind domain with non-torsion class group. Let $P$ be a prime of infinite order in said group. Let $B = \bigcap_{Q \in \text{Max} A, Q \neq P} A_Q$. Then I 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 $A$. – Neil Epstein Mar 20 '15 at 15:49