In algebraic geometry, a finite morphism between two affine varieties
is a dense regular map which induces isomorphic inclusion
between their coordinate rings, such that
[1] This definition can be extended to the quasi-projective varieties, such that a regular map
between quasiprojective varieties is finite if any point
has an affine neighbourhood V such that
is a finite map (in view of the previous definition, because it is between affine varieties).
[2] 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.
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.
is a finite morphism since
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.