A dominant affine map factors finitely over relative affine space after shrinking the base

Given: The objects and hypotheses in the statement.

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).

Let $k$ be a field and let $A$ be a nonzero finite-type $k$-algebra. Then there exist algebraically independent elements $z_1,\dots ,z_d\in A$ such that $A$ is a module-finite algebra over the polynomial ring $k[z_1,\dots ,z_d]$. (Noether normalisation yields module finiteness over a polynomial subring).

Assume the Axiom of Choice. Let $X$ be a classical affine variety over an algebraically closed field $k$, and let $\bar{f}\in k[X]$. Put $U=D_X(\bar{f})$. A function $\phi :U\to k$ is called regular on $U$ if there exist finitely many pairs $({\bar{g}}_i,{\bar{h}}_i)$ in $k[X]\times k[X]$ such that $U={\bigcup }_{i=1}^r{D}_X({\bar{h}}_i)$ and $\phi (x)=\frac{{\bar{g}}_i(x)}{{\bar{h}}_i(x)}\text{for every }x\in D_X({\bar{h}}_i)\mathrm{.}$ Write ${\mathcal{O}}_X(U)$ for the ring of regular functions on $U$. Then evaluation induces a ring isomorphism $k[X{]}_{\bar{f}}\stackrel{\sim }{\to }{\mathcal{O}}_X(U)\mathrm{.}$ If $U=\mathrm{\varnothing }$, both sides are the zero ring. (Regular functions on a principal open are the principal localization of the coordinate ring).