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Finite morphism

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In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes

Contents

V i = 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 → A i ,

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 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.

For example, for any field k, the morphism from the affine line A1 over k to itself given by x ↦ x2 is finite. (Indeed, the polynomial ring k[x] is finitely generated as a module over k[y] by y ↦ x2, with generators 1 and x.) 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].)

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 corresponding to the closed subscheme.
  • Finite morphisms are closed, hence (because of their stability under base change) proper. This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
  • Finite morphisms have finite fibers (that is, they are quasi-finite). 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. 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.
  • Finite morphisms are both projective and affine.

    • Morphisms of finite type

    For a homomorphism A → B 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.

    The analogous notion in terms of schemes is: a morphism f: X → Y 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.

    References

    Finite morphism Wikipedia
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