Finite morphism

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 ${\displaystyle y\in Y}$ 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: X → Y 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 A_i, and the restriction of f to U_i, which induces a ring homomorphism

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

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, ${\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])}$ 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: X → Y is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×_YZ → 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 ⊗_BC 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

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

References

• Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS . 28: 5–255. doi:10.1007/bf02684343. MR   0217086.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS . 32: 5–361. doi:10.1007/bf02732123. MR   0238860.
• Hartshorne, Robin (1977), Algebraic Geometry , Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN   978-0-387-90244-9, MR   0463157
• Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer Science. doi:10.1007/978-3-642-37956-7. ISBN   978-0-387-97716-4.