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
such that for each i,
is an open affine subscheme Spec A_{i}, and the restriction of f to U_{i}, which induces a ring homomorphism
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, 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 A^{1} − 0 into A^{1} 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[x_{1}, ..., x_{n}] 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: 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 A^{n} 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 P^{n}, where n is the dimension of X.
See also
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