Finite morphism

In algebraic geometry, 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 A_i, and the restriction of f to U_i, which induces a ring homomorphism

B_{i}\rightarrow A_{i},

makes A_i a finitely generated module over B_i.^[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, 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 a_i ⊗ 1, where a_i 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.^[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: 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.^[7]

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, ..., x_n] 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 V_i = Spec A_i such that f⁻¹(V_i) has a finite covering by affine open subschemes U_ij = Spec B_ij with B_ij an A_i-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.

Notes

↑ Hartshorne (1977), section II.3.
↑ Stacks Project, Tag 01WG .
↑ Stacks Project, Tag 01WG .
↑ Stacks Project, Tag 01WG .
↑ Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
↑ Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
↑ Stacks Project, Tag 01WG .

References

External links

The Stacks Project Authors, The Stacks Project

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