I'm trying to work out a proof of the upper semi-continuity of the dimension of the fibers of a morphism as it appears in Liu's book, exercise 4.4.6.
I'm well aware of the classical proof using Zariski Main Theorem, that goes along those lines, first reduce to the affine case, and then if $X \to Y$ happens to have a fiber of dimension say $d$, then one gets (by reducing $Y$ if necessary) a map $\mathbb{A}^d_Y \to X$ (over $Y$) that has a finite fiber over some $x$ lying over $y$, reducing $X$ further that means that $\mathbb{A}^d_Y\times_X U \to U$ has finite fibers, but for $u$ in $U$ that means that the dimension at $u$ of $X_{f(u)}$ is at most $d$.
Liu seems to take a different approach, although he places the exercise in his chapter about ZMT and this confuses me a bit.
His approach goes as follows, take $X,Y$ locally noetherian, irreducible, and $X\to Y$ a finite type morphism, and $y \in Y$.
We want to prove that the dimension of $X_y$ is larger than the dimension of the generic fiber.
He wants to do an induction over $\dim O_{Y,y}+ \dim X_y$, so he takes $x$ in the fiber, such that $\dim O_{X_y,x}=\dim X_y$, the case where the dimension of the fiber is 0 being ZMT, then we may assume that the fiber is at least 1 dimensional.
By reducing $X$ (and taking it affine) one may find $b\in O_X(X)$ such that $b$ doesn't vanish on the irreducible components of $X_y$, and taking an irreducible component say $Z$, of $V(b)$ that is not included in $X_y$ and $x\in V(b)$, if we take $y'$ the generic point of the image of $Z$ in $Y$ he then claims that
- $\dim O_{Y,y'} < \dim O_{Y,y}$, this is clear as $y'$ specializes to $y$
- $\dim X_{y'}-\dim X_{y} \leq \dim Z_{y'}-\dim Z_{y}$
And I'm not sure I understand why 2 is true. I think the reason for this (I'm not too sure about what follows) is $Z_{y'}$ is included in something cut out by $b=0$ in $X_{y'}$ and thus is at best a hypersurface, and thus $$\dim X_{y'}-\dim Z_{y'} \geq 1$$ where as $Z_y$ should be exactly a hypersurface of $X_y$ as $b$ is regular in $O_{X_y}(X_y)$ and thus the irreducible components of $V(b)$ should be of dimension $1$ (but it is not clear why $Z_y$ remains an irreducible component of $V(b)$ in $X_y$) that should give $$\dim X_{y}-\dim Z_{y} = 1$$ and the two inequalities combined should give the result.
I'm having a hard time making this precise, and also I'm a bit puzzled by the fact that this does not use ZMT at all. I'm thinking 2 should follow from some form of ZMT but I've failed to find a way to derive it from ZMT.
Is the approach (for proving 2) that I outlined the correct one? Or am I missing something? And if it's the correct approach, how can I make this rigorous?
Addendum:
Ok, I think I understood the idea of the proof, maybe this is what Liu has in mind.
The theorem is clear in several cases, if the fiber $X_y$ is of dimension 0 (this is ZMT), or if $y$ is itself the generic point of $Y$ as there is nothing to prove.
If we are not in those cases we cut out a hypersurface in $X$, several things may happen, either the dimension of the fiber intersected with the hypersurface strictly decreases and by continuing the process we ultimately get to 0 dimensional fiber, or the dimension of the image of that hypersurface in $Y$ decreases, and again ultimately we get to $Y$ being a point and we're done, but it may happen that none of this happens (and also we may need to ping pong between the two phenomena) this is the worst case when neither the fiber, nor the image of $Z$ in $Y$ decreases, but that means that an irreducible component of the fiber is contained in $Z$, and in that case we certainly have $\dim(Z) < \dim (X)$, so ultimately again, we have an irreducible component of the fiber that is equal to the whole of $Z$ with $Z$ mapping dominantely to $Y$ that means that $Y$ has $y$ as generic point and we have the result by the other obvious case.
Of course I treated as each case "repeated itself" but in practice everytime we cut out by a hypersurface we are in one the 3 cases, and after sufficient step we should reach one "terminal case".
Maybe this implies Liu's 2 inequality (as a "global" measure of the size of the case, but I feel like this would be closer to $\dim O_{Y,y} +\dim X_y + \text{codim}(X_y, Y)$, but I'm sure I am missing something)