Proper quasi-finite classical morphisms are finite

Apply [F1], retaining separatedness and finite type from the properness assumption. The finite map $g$ is separated: over $\mathrm{Spec}A\subseteq Y$, write its inverse image as $\mathrm{Spec}B$, where $B$ is a finite $A$-algebra; its relative diagonal is closed because the multiplication map $B{\otimes }_{A}B\twoheadrightarrow B$ is surjective. Thus the graph of $j$ is closed in $X{\times }_{Y}Z$, as it is the inverse image of this diagonal under $(x,z)\mapsto (j(x),z)$.

The projection $X{\times }_{Y}Z\to Z$ is a base change of the universally closed map $f$, so the image of the graph is closed in $Z$. This image is $j(X)$. It is open by [F1], hence open and closed. The open immersion identifies $X$ with this subvariety; its inclusion is also a closed immersion. On any affine open $V=\mathrm{Spec}A$ of $Y$, $g^{-1}(V)=\mathrm{Spec}B$ with $B$ finite over $A$. The open-and-closed subset $j(X)\cap g^{-1}(V)$ is affine: its characteristic function is locally the regular constants $1$ and $0$, which glue to an idempotent $e\in B$; the subset is $D(e)=V(1-e)$ and has coordinate ring $B\mathrm{/}(1-e)$. This is a quotient of a finite $A$-module, hence finite over $A$. Therefore $f$ is finite on every affine target open, which proves the assertion.

For the stated projective case, write the morphism as a closed subvariety of $Y\times {\mathbf{P}}_k^N$ followed by projection. After any classical base change $Y^{\mathrm{\prime }}\to Y$ this remains a closed subvariety of $Y^{\mathrm{\prime }}\times {\mathbf{P}}_k^N$, so [F3] makes its projection closed. The morphism is separated (its relative diagonal is given by the projective cross-product equations) and of finite type (its finite standard affine charts have finite-type coordinate rings). With finite fibres it therefore satisfies all the proper quasi-finite hypotheses of steps 1.1–2.1 and is finite. This proves Chevalley's projective finite-fibre case without assuming algebraic closure of the image model or an affine source.