Finite fibres do not imply a module-finite map

Given: The objects and hypotheses in the statement refuted, for the explicit witness below.

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 affine classical algebraic sets $X,Y$, call a morphism $f:X\to Y$ module-finite if $k[X]$, via pullback, is a finitely generated $k[Y]$-module. 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. 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. (Module-finite affine maps for the quasi-finite comparison).

For a morphism $f:X\to Y$ of classical varieties and a closed point $y\in Y$, let $X_y=f^{-1}(y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_y=-\mathrm{\infty }$ if the fibre is empty. On affine charts $V\subseteq Y$ containing $y$ and $U\subseteq f^{-1}(V)$, writing $A=k[V]$ and $B=k[U]$, the fibre chart has coordinate ring $B\mathrm{/}\sqrt{{\mathfrak{m}}_{y}B}$. Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U\to V$. 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. (Reduced closed-point fibres and their dimension).