Finite morphism

In algebraic geometry, a finite morphism between two affine varieties <math>X, Y</math> is a dense regular map which induces isomorphic inclusion <math>k\left[Y\right]\hookrightarrow k\left[X\right]</math> between their coordinate rings, such that <math>k\left[X\right]</math> is integral over <math>k\left[Y\right]</math>. This definition can be extended to the quasi-projective varieties, such that a regular map <math>f\colon X\to Y</math> between quasiprojective varieties is finite if any point <math>y\in Y</math> has an affine neighbourhood V such that <math>U=f^{-1}(V)</math> is affine and <math>f\colon U\to V</math> 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

:<math>V_i = \mbox{Spec} \; B_i</math>

such that for each i,

:<math>f^{-1}(V_i) = U_i</math>

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

:<math>B_i \rightarrow A_i,</math>

makes A_i a finitely generated module over B_i (in other words, a finite B_i-algebra). 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, <math>\text{Spec}(k[t,x]/(x^n-t)) \to \text{Spec}(k[t])</math> is a finite morphism since <math>k[t,x]/(x^n-t) \cong k[t]\oplus k[t]\cdot x \oplus\cdots \oplus k[t]\cdot x^{n-1}</math> as <math>k[t]</math>-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 a_i ⊗ 1, where a_i 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 (section of the ideal sheaf) corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper. 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.