Let $f: X \to Y$ a dominant morphism of varieties
over a field $k$. My 'varieties' are by definition irreducible
and let denote as $g_X \in X$ and $g_Y \in Y$ their generic points. Dominant
means that $f(g_X)=g_Y$. Let $n= \dim(X), m= \dim(Y)$.
Then it is known that the transcendence degree of function fields
of varieties coinside with their dimensions. Therefore
$\operatorname{trdeg}_k \ K(X) =\dim(X)=n$ and $\operatorname{trdeg}_k \ K(Y) =\dim(Y)$.
I want to check that the dimension of the
generic fiber $X_{g_Y} = f^{-1}(g_Y)$ is
$e= n-m$.
My 'proof' has some gaps I would like to plug. Assume that
both varieties are affine: $X= \operatorname{Spec}(A), Y=\operatorname{Spec}(R)$. The map $f$
is dominant, therefore $R \to A$ is injective.
The affine ring of generic fiber $X_{g_Y}$
is $A \otimes_R K(Y)$.
Since $g_X \in X_{g_Y} $ there exist a map
$A \otimes_R K(Y) \to K(X)= \operatorname{Frac}(A)$ und therefore,
since $K(X)$ is a field, we get the induced inclusion of fields
$K(A \otimes_R K(Y)) \subset K(X)$. $K(A \otimes_R K(Y))$ is the function field of generic fiber $X_{g_Y}$.
Question: what do we know about the field extension
$K(A \otimes_R K(Y)) \subset K(X)$? which transcendence degree
does it have?
My idea is obvious: if I can prove that
$\operatorname{trdeg}_{K(A \otimes_R K(Y)} K(X)= m$ then I can
use additivivity formula for transcendence degrees of field extensions
to show that $\operatorname{trdeg}_k K(A \otimes_R K(Y))=n-m$ because of
$\operatorname{trdeg}_k \ K(X) =n$ and additivity for tower of extensions $k \subset K(A \otimes_R K(Y) \subset K(X)$. But I don't know
how I can figure out that
$\operatorname{trdeg}_{K(A \otimes_R K(Y)} K(X)= m$.
Best Answer
As Youngsu pointed out, the fields $K(A\otimes_RK(Y))$ and $K(X)$ are indeed equal. To see this, note that $A\otimes_RK(Y)$ is the localisation of the domain $A$ at its multiplicative subset $R\setminus\{0\}\subseteq A$, and $K(A\otimes_RK(Y))$ is obtained from $A\otimes_RK(Y)$ by localising at the remaining non-zero elements. By transitivity of localisations, this shows that $K(A\otimes_RK(Y))$ can also be written as the localisation of $A$ at all its non-zero elements, which is $K(X)$.
So what goes wrong here? The problem is that the generic fibre $X_{g_Y}$ is usually not a variety over $k$! Suppose, for example, that $X=Y$ and $f\colon X\rightarrow X$ the identity. Then the generic fibre is $\operatorname{Spec} K(X)$, and $K(X)$ is usually not of finite type as a $k$-algebra, unless it is a finite field extension of $k$ (by Hilbert's Nullstellensatz), i.e., unless $X$ is $0$-dimensional.
However, $X_{g_Y}$ is a variety* over $K(Y)$ (can you show that?), and it is indeed true that $X_{g_Y}$ is $(n-m)$-dimensional as a $K(Y)$-variety! So instead of $\operatorname{trdeg}_kK(A\otimes_RK(Y))=n-m$, you need to show that $$\operatorname{trdeg}_{K(Y)}K(A\otimes_RK(Y))=n-m\,.$$ Now your idea of using transitivity of transcendence degrees works: We've seen above that $K(A\otimes_RK(Y))=K(X)$, and $\operatorname{trdeg}_{K(Y)}K(X)=\operatorname{trdeg}_kK(X)-\operatorname{trdeg}_kK(Y)=n-m$.
* Usually, $K(Y)$ won't be algebraically closed, but I think you're working with the scheme-theoretic definition of varieties anyway.