In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]} between their coordinate rings, such that k [ X ] {\displaystyle k\left[X\right]} is integral over k [ Y ] {\displaystyle k\left[Y\right]} . This definition can be extended to the quasi-projective varieties, such that a regular map f : X → Y {\displaystyle f\colon X\to Y} between quasiprojective varieties is finite if any point y ∈ Y {\displaystyle y\in Y} has an affine neighbourhood V such that U = f − 1 ( V ) {\displaystyle U=f^{-1}(V)} is affine and f : U → V {\displaystyle f\colon 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 {\displaystyle V_{i}={\mbox{Spec}}\;B_{i}}such that for each i,
f − 1 ( V i ) = U 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
B i → A i , {\displaystyle B_{i}\rightarrow A_{i},}makes Ai a finitely generated module over Bi (in other words, a finite Bi-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, Spec ( k [ t , x ] / ( x n − t ) ) → Spec ( k [ t ] ) {\displaystyle {\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])} is a finite morphism since k [ t , x ] / ( x n − t ) ≅ k [ t ] ⊕ k [ t ] ⋅ x ⊕ ⋯ ⊕ k [ t ] ⋅ x n − 1 {\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 k [ t ] {\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 ×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 (section of the ideal sheaf) 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
See also
Notes
- 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.
References
Shafarevich 2013, p. 60, Def. 1.1. - 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 ↩
Shafarevich 2013, p. 62, Def. 1.2. - 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 ↩
Hartshorne 1977, Section II.3. - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 https://mathscinet.ams.org/mathscinet-getitem?mr=0463157 ↩
Stacks Project, Tag 01WG. http://stacks.math.columbia.edu/tag/01WG ↩
Stacks Project, Tag 01WG. http://stacks.math.columbia.edu/tag/01WG ↩
Stacks Project, Tag 01WG. http://stacks.math.columbia.edu/tag/01WG ↩
Grothendieck, EGA IV, Part 4, Corollaire 18.12.4. ↩
Grothendieck, EGA IV, Part 3, Théorème 8.11.1. ↩
Stacks Project, Tag 01WG. http://stacks.math.columbia.edu/tag/01WG ↩