= Proper affine morphisms are finite
{title2=$\operatorname{Spec}B\to\operatorname{Spec}A\text{ proper}\Longrightarrow B\text{ finite over }A$}
For a finite-type affine morphism of Noetherian schemes, apply its valuative lifting property to each domain quotient $B/\mathfrak q$ for a minimal prime and all valuation rings of its fraction field containing the image of $A$. The lift forces $B/\mathfrak q$ into every such valuation ring, hence makes it integral over that image. For each $b\in B$, multiply the finitely many resulting monic equations. Its value at $b$ lies in the nilradical. The nilradical of a Noetherian ring is nilpotent, so a power of this monic polynomial annihilates $b$. Thus $B$ is integral over $A$. Finitely many algebra generators, each satisfying a monic equation, have their powers reduced to finitely many monomials, proving that $B$ is a finite $A$-module. Nilpotents cannot be ignored in this proof.
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