Finite morphism

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

[math]\displaystyle{ V_i = \mbox{Spec} \; B_i }[/math]

such that for each i,

[math]\displaystyle{ f^{-1}(V_i) = U_i }[/math]

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

[math]\displaystyle{ B_i \rightarrow A_i, }[/math]

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.

For example, for any field k, [math]\displaystyle{ \text{Spec}(k[t,x]/(x^n-t)) \to \text{Spec}(k[t]) }[/math] is a finite morphism since [math]\displaystyle{ k[t,x]/(x^n-t) \cong k[t]\oplus k[t]\cdot x \oplus\cdots \oplus k[t]\cdot x^{n-1} }[/math] as [math]\displaystyle{ k[t] }[/math]-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. 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.

See also

• Glossary of algebraic geometry
• Finite algebra

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. http://www.numdam.org/articles/PMIHES_1966__28__5_0. 
• 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. http://www.numdam.org/articles/PMIHES_1967__32__5_0. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 
• Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer Science. doi:10.1007/978-3-642-37956-7. ISBN 978-0-387-97716-4. https://link.springer.com/book/10.1007/978-3-642-37956-7. 

External links

• The Stacks Project Authors, The Stacks Project, http://stacks.math.columbia.edu/