Finite morphism

In algebraic geometry, a finite morphism between two affine varieties $$X, Y$$ is a dense regular map which induces isomorphic inclusion $$k\left[Y\right]\hookrightarrow k\left[X\right]$$ between their coordinate rings, such that $$k\left[X\right]$$ is integral over $$k\left[Y\right]$$. This definition can be extended to the quasi-projective varieties, such that a regular map $$f\colon X\to Y$$ 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 $$f\colon U\to V$$ 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
 * $$V_i = \mbox{Spec} \; B_i$$

such that for each i,


 * $$f^{-1}(V_i) = U_i$$

is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism


 * $$B_i \rightarrow A_i,$$

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, $$\text{Spec}(k[t,x]/(x^n-t)) \to \text{Spec}(k[t])$$ is a finite morphism since $$k[t,x]/(x^n-t) \cong k[t]\oplus k[t]\cdot x \oplus\cdots \oplus k[t]\cdot x^{n-1}$$ 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

 * The composition of two finite morphisms is finite.
 * Any base change of a finite morphism f: X → Y is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is finite. This corresponds to the following algebraic statement: if A and C are (commutative) B-algebras, and A is finitely generated as a B-module, then the tensor product A ⊗B C is finitely generated as a C-module. Indeed, the generators can be taken to be the elements ai ⊗ 1, where ai are the given generators of A as a B-module.
 * Closed immersions are finite, as they are locally given by A → A/I, where I is the ideal corresponding to the closed subscheme.
 * Finite morphisms are closed, hence (because of their stability under base change) proper. This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
 * Finite morphisms have finite fibers (that is, they are quasi-finite). This follows from the fact that for a field k, every finite k-algebra is an Artinian ring. A related statement is that for a finite surjective morphism f: X → Y, X and Y have the same dimension.
 * By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite. This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is Noetherian.
 * Finite morphisms are both projective and affine.