Equal dimension is equivalent to generic quasi-finiteness

Given: The objects and hypotheses in the statement.

For a dominant morphism $f:X\to Y$ between irreducible classical varieties, there is a nonempty open $U\subseteq Y$, contained in $f(X)$, such that every $X_y$ with $y\in U$ is nonempty and has pure dimension $r=\dim X-\dim Y$. 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. (Fibres have pure expected dimension over a dense open).

A classical variety $X$ has $\dim X\le 0$ if and only if its underlying set is finite. The empty set is included. A nonempty irreducible variety of dimension zero is one point. 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. (Zero-dimensional varieties are finite sets).

A morphism $f:X\to Y$ of classical varieties is quasi-finite if every closed-point fibre $X_y$ is a finite set; empty fibres are allowed. Classical morphisms here are of finite type: for an affine target chart and an affine source chart above it, any finite set of $k$-algebra generators of the source ring also generates it over the target ring. The inverse image has a finite affine cover because it is an open of a Noetherian variety. 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. (Quasi-finite classical morphisms).

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 $k\subseteq K\subseteq L$ be a tower of field extensions. Assume that ${\mathrm{trdeg}}_{k}K$ and ${\mathrm{trdeg}}_{K}L$ are finite. Then ${\mathrm{trdeg}}_{k}L={\mathrm{trdeg}}_{k}K+{\mathrm{trdeg}}_{K}L\mathrm{.}$ (Transcendence degree is additive in finite towers).

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. 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. (Dimension equals transcendence degree).