Finite morphism

In algebraic geometry, a finite morphism between two affine varieties $X, Y$ is a dense regular map which induces isomorphic inclusion $k [Y] ↪ k [X]$ between their coordinate rings, such that $k [X]$ is integral over $k [Y]$ . This definition can be extended to the quasi-projective varieties, such that a regular map $f : X \to Y$ 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 $f : U \to V$ 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} \to A_{i},

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

Notes

↑ Shafarevich 2013, p. 60, Def. 1.1.
↑ Shafarevich 2013, p. 62, Def. 1.2.
↑ Hartshorne 1977, Section II.3.
1 2 3 4 Stacks Project, Tag 01WG.
↑ Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
↑ Grothendieck, EGA IV, Part 3, Théorème 8.11.1.

[FOOTNOTEShafarevich201360Def._1.1-1] Shafarevich 2013, p. 60, Def. 1.1.

[FOOTNOTEShafarevich201362Def._1.2-2] Shafarevich 2013, p. 62, Def. 1.2.

[FOOTNOTEHartshorne1977Section_II.3-3] Hartshorne 1977, Section II.3.

[iXBUK-4] 1 2 3 4 Stacks Project, Tag 01WG.

[5] Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.

[6] Grothendieck, EGA IV, Part 3, Théorème 8.11.1.

Finite morphism

Definition by schemes

Properties of finite morphisms

See also

Notes

References