Finite morphism

Summary

In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over .[1] This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point like has an affine neighbourhood V such that is affine and is a finite map (in view of the previous definition, because it is between affine varieties).[2]

Definition by SchemesEdit

A morphism f: XY of schemes is a finite morphism if Y has an open cover by affine schemes

 

such that for each i,

 

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

 

makes Ai a finitely generated module over Bi.[3] One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine open 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.

Properties of finite morphismsEdit

  • The composition of two finite morphisms is finite.
  • Any base change of a finite morphism f: XY is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y ZZ 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 AB 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 AA/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: XY, 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: XY 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 alsoEdit

NotesEdit

  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. ^ Stacks Project, Tag 01WG.
  6. ^ Stacks Project, Tag 01WG.
  7. ^ Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
  8. ^ Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
  9. ^ Stacks Project, Tag 01WG.

ReferencesEdit

External linksEdit

  • The Stacks Project Authors, The Stacks Project