Finite morphism

In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]} between their coordinate rings, such that k [ X ] {\displaystyle k\left[X\right]} is integral over k [ Y ] {\displaystyle k\left[Y\right]} .^[1] This definition can be extended to the quasi-projective varieties, such that a regular map f : X → Y {\displaystyle f\colon X\to Y} between quasiprojective varieties is finite if any point y ∈ Y {\displaystyle y\in Y} has an affine neighbourhood V such that U = f − 1 ( V ) {\displaystyle U=f^{-1}(V)} is affine and f : U → V {\displaystyle f\colon U\to V} is a finite map (in view of the previous definition, because it is between affine varieties).^[2]

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 = Spec B i {\displaystyle V_{i}={\mbox{Spec}}\;B_{i}}

such that for each i,

f − 1 ( V i ) = U i {\displaystyle f^{-1}(V_{i})=U_{i}}

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

B i → A i , {\displaystyle B_{i}\rightarrow A_{i},}

makes A_i a finitely generated module over B_i (in other words, a finite B_i-algebra).^[3] 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.^[4]

For example, for any field k, Spec ( k [ t , x ] / ( x n − t ) ) → Spec ( k [ t ] ) {\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])} is a finite morphism since k [ t , x ] / ( x n − t ) ≅ k [ t ] ⊕ k [ t ] ⋅ x ⊕ ⋯ ⊕ k [ t ] ⋅ x n − 1 {\displaystyle 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 ] {\displaystyle 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 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.^[4] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
• Finite morphisms have finite fibers (that is, they are quasi-finite).^[4] 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.^[5] 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.^[6]
• Finite morphisms are both projective and affine.^[4]

See also

• Glossary of algebraic geometry
• Morphism of finite type

Notes

References