Theorem. Let f : X —> Y be a proper morphism of varieties and let y ∈ Y with f^{-1}(y) finite. Then there exists a neighborhood V of y in Y such that f^{-1}(V) —> V is finite.

If X is quasi-projective, then there is a simple proof: Choose an affine open U of X containing f^{-1}(y); this uses X quasi-projective. Using properness of f, find an affine open V ⊂ Y such that f^{-1}V ⊂ U. Then f^{-1}V = V x_Y U is affine as Y is separated. Hence f^{-1}V —> V is a proper morphism of affines varieties. Such a morphism is finite, see Lemma Tag 01WM for an elementary argument.

I do not know a truly simple proof for the general case. (Ravi explained a proof to me that avoids most cohomological machinery, but unfortunately I forgot what the exact method was; it may even be one of the arguments I list below.) Here are some different approaches.

(A) One can give a proof using cohomology and the theorem on formal functions, see Lemma Tag 020H.

Let ZMT be Grothendieck’s algebraic version of Zariski’s main theorem, see Theorem Tag 00Q9.

(B) One can prove the result using ZMT and etale localization. Namely, one proves that given any finite type morphism X —> Y with finite fibre over y, there is after etale localization on Y, a decomposition X = U ∐ W with U finite over Y and the fibre W_y empty (see Section Tag 04HF). In the proper case it follows that W is empty after shrinking Y. Finally, etale descent of the property “being finite” finishes the argument. This method proves a general version of the result, see Lemma Tag 02LS.

(C) A mixture of the above two arguments using ZMT and a characterization of affines:

1. Show that after replacing Y by a neighborhood of y we may assume that all fibers of f are finite. This requires showing that dimensions of fibres go up under specialization. You can prove this using generic flatness and the dimension formula (as in Eisenbud for example) or using ZMT.
2. Let X’ —> Y be the normalization of Y in the function field of X. Then X’ —> Y is finite and X’ and X are birational over Y. Finiteness of X’ over Y requires finiteness of integral closure of finite type domains over fields, which follows from Noether normalization + epsilon.
3. Let W ⊂ X x_Y X’ be the closure of the graph of the birational rational map from X to X’. Then W —> X is finite and birational and W —> X’ is proper with finite fibres and birational.
4. Using ZMT one shows that W —> X’ is an isomorphism. Namely, a corollary of ZMT is that separated quasi-finite birational morphisms towards normal varieties are open immersions.
5. Now we have X’ —> X —> Y with the first arrow finite birational and the composition finite too. After shrinking Y we may assume Y and X’ are affine. If X is affine, then we win as O(X) would be a subalgebra ofa finite O(Y)-algebra.
6. Show that X is affine because it is the target of a finite surjective morphism from an affine. Usually one proves this using cohomology. The Noetherian case is Lemma Tag 01YQ (this uses less of the cohomological machinery but still uses the devissage of coherent modules on Noetherian schemes). In fact, the target of a surjective integral morphism from an affine is affine, see Lemma Tag 05YU.