Finite morphism explained

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} = Spec B _{i}

such that for each i,

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,}

makes A_i a finitely generated module over B_i. 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.^[1]

For example, for any field k,

Spec (k[t,x]/(x ^{n-t))} \to Spec (k[t]) is a finite morphism since k[t,x]/(x ^{n-t)} \cong k[t] ⊕ k[t] ⋅ x ⊕ … ⊕ k[t] ⋅ 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 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 corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper.^[2] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
• Finite morphisms have finite fibers (that is, they are quasi-finite).^[3] 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.^[4] 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.^[5]
• Finite morphisms are both projective and affine.^[6]

See also

• Glossary of algebraic geometry
• Finite algebra

References

• • • • Book: Shafarevich, Igor R. . Igor Shafarevich. Basic Algebraic Geometry 1 . 2013. . 10.1007/978-3-642-37956-7 . 978-0-387-97716-4.

Notes and References

1. .
2. .
3. .
4. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
5. Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
6. .