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}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6cdbfbe67303c0982d838a8432e512ade9a20afe)
such that for each i,
![{\displaystyle f^{-1}(V_{i})=U_{i}}](//wikimedia.org/api/rest_v1/media/math/render/svg/58b665ff5fdea974d8343ddae226537112e27c2d)
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},}](//wikimedia.org/api/rest_v1/media/math/render/svg/fb936dc815574f3c3f33fe2f4c92438c3f1c7daf)
makes Ai a finitely generated module over Bi. 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,
is a finite morphism since
as
-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.
- 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 ai ⊗ 1, where ai 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.[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: 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.[7] 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.[8]
- Finite morphisms are both projective and affine.[9]
Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
- 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.