Finite morphism

In algebraic geometry, 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.[1] 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.[2]

For example, for any field k, is a finite morphism since as -modules. Geometrically, this is obviously finite since this 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: 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.[3] 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: 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.[5] 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.[6]
  • Finite morphisms are both projective and affine.[7]

Morphisms of finite type

For a homomorphism AB of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ring A[x1, ..., xn] is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0. Another example of a finite-type morphism which is not finite is .

The analogous notion in terms of schemes is: a morphism f: XY of schemes is of finite type if Y has a covering by affine open subschemes Vi = Spec Ai such that f−1(Vi) has a finite covering by affine open subschemes Uij = Spec Bij with Bij an Ai-algebra of finite type. One also says that X is of finite type over Y.

For example, for any natural number n and field k, affine n-space and projective n-space over k are of finite type over k (that is, over Spec k), while they are not finite over k unless n = 0. More generally, any quasi-projective scheme over k is of finite type over k.

The Noether normalization lemma says, in geometric terms, that every affine scheme X of finite type over a field k has a finite surjective morphism to affine space An over k, where n is the dimension of X. Likewise, every projective scheme X over a field has a finite surjective morphism to projective space Pn, where n is the dimension of X.

See also

Notes

  1. Hartshorne (1977), section II.3.
  2. Stacks Project, Tag 01WG.
  3. Stacks Project, Tag 01WG.
  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.
  7. Stacks Project, Tag 01WG.

References

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