I have a question about a claim I found in Liu's book Algebraic Geometry and Arithmetic Curves in the Proof of Theorem 8.3.26 (b) page 356:
In (b) uses the notation of a "center". This can be looked up on previous pages:
By a center $x \in X$ of valaution $\nu$ of $K(X)$ Liu means the existence of a morphism $\operatorname{Spec} \ O_{\nu} \to \operatorname{Spec} \ O_{X,x}$ extending canonical inclusion of generic point $Spec \ K(X) \to X$. $O_{\nu}$ is the DVR with maximal ideal $m_{\nu}$ induced by $\nu$.
Also, the proof uses following assertion which we consider as black box:
($\alpha$) There exists a blowing-up morphism $X' \to X$ such that the center $x'$ of $\nu$ in $X'$ verifies $\operatorname{trdeg}_{k(x)} k(x') ≥ n$.
The relevant part I not understand is:
Let $Y \to X'$ be the normalization. This is a finite morphism because $X'$ is Nagata (Proposition 2.29(b); we can also accept it as black box). Hence $\nu$ admits a center $y \in Y$ which is a point lying above $x'$.
QUESTION: Why finiteness of $Y \to X'$ allows to lift the center $x'$ with respect to $\nu$ to $Y$?
I think the problem can be translated to commutative algebra as follows:
The center is: if $\nu$ is a non-trivial valuation of $K(X)$ and has $x \in X$ center, this means simply that $O_{\nu}$ dominates $O_{X,x}$, i.e. we have $O_{X,x} \subset O_{\nu}$ and $m_x \subset m_{\nu}$ for unique maximal ideals (equivalently: $m_{\nu} \cap O_{X,x}=m_x$.
Assume $\nu$ has center in $x' \in X'$, that is $O_{x',X'} \subset O_{\nu}$ and $O_{\nu}$ dominates $O_{x',X'} $. Let $y_1,...,y_m$ the primages of $x'$ with respect the normalization map. By Nagata argument the normalization map was finite and thus we have finitely many preimages.
futhermore for every $y_i$ the induced maps between local rings $O_{X',x'} \to O_{Y,y_i}$ are finite.
That is the question is why there exist a $y_i$ such that $O_{x',X'} \subset O_{\nu}$ extends to $O_{y_i,Y} \subset O_{\nu}$?