In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over. This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point y\in Y has an affine neighbourhood V such that U=f^{-1}(V) is affine and is a finite map (in view of the previous definition, because it is between affine varieties).

Definition by schemes

A morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi. One also says that X is finite over Y. In fact, f is finite if and only if for every open affine subscheme V = Spec B in Y, the inverse image of V in X is affine, of the form Spec A, with A a finitely generated B-module. For example, for any field k, is a finite morphism since as k[t]-modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of A1 − 0 into A1 is not finite. (Indeed, the Laurent polynomial ring k[y, y−1] is not finitely generated as a module over k[y].) This restricts our geometric intuition to surjective families with finite fibers.

Properties of finite morphisms