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

V_i = \mbox{Spec} \; B_i

such that for each i,

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 \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, \text{Spec}(k[t,x]/(x^n-t)) \to \text{Spec}(k[t]) is a finite morphism since 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]-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.[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

Notes

  1. ^ Shafarevich 2013, 60.
  2. ^ Shafarevich 2013, 62.
  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