Module-finite affine maps for the quasi-finite comparison

For affine classical algebraic sets $X,Y$, a morphism $f:X\to Y$ induces a unital $k$-algebra homomorphism $f^{\ast }:k[Y]\to k[X]$ by pullback (Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms, The coordinate ring of a classical affine algebraic set). Call $f$ module-finite when $k[X]$ is finitely generated as a $k[Y]$-module for the action $a\cdot b=f^{\ast }(a)b$ (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras). This is the affine module criterion. Empty affine sets are allowed, with zero coordinate ring; the definition does not assert a global affine-preimage criterion for arbitrary varieties.

Work over a fixed algebraically closed field $k$, with the Axiom of Choice inherited from the pullback dictionary (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.