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, ${\text{Spec}}(k[t,x]/(x^{n}-t))\to {\text{Spec}}(k[t])$ 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 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-module unless A = 0 or n = 0. Another example of a finite-type morphism which is not finite is $\mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)$ .

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

Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255. doi:10.1007/bf02684343. MR 0217086.
Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

External links

The Stacks Project Authors, The Stacks Project

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[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.