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Finite morphism

## Summary

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

## Definition by Schemes

A morphism f: XY of schemes is a finite morphism if Y has an open cover by affine schemes

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

such that for each i,

${\displaystyle f^{-1}(V_{i})=U_{i}}$

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

${\displaystyle B_{i}\rightarrow A_{i},}$

makes Ai a finitely generated module over Bi.[3] One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine open 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, ${\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])}$  is a finite morphism since ${\displaystyle k[t,x]/(x^{n}-t)\cong k[t]\oplus k[t]\cdot x\oplus \cdots \oplus k[t]\cdot x^{n-1}}$  as ${\displaystyle k[t]}$ -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: XY is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y ZZ 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 AB 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 AA/I, where I is the ideal corresponding to the closed subscheme.
• Finite morphisms are closed, hence (because of their stability under base change) proper.[5] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
• Finite morphisms have finite fibers (that is, they are quasi-finite).[6] 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: XY, 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.[7] This had been shown by Grothendieck if the morphism f: XY is locally of finite presentation, which follows from the other assumptions if Y is Noetherian.[8]
• Finite morphisms are both projective and affine.[9]

## Notes

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