Finite morphism
In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over .[1] This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point like has an affine neighbourhood V such that is affine and is a finite map (in view of the previous definition, because it is between affine varieties).[2]
Definition by Schemes[]
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 Ai, and the restriction of f to Ui, which induces a ring homomorphism
makes Ai a finitely generated module over Bi.[3] 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.[4]
For example, for any field k, is a finite morphism since as -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 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.[5] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra.
- Finite morphisms have finite fibers (that is, they are quasi-finite).[6] 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.[7] 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.[8]
- Finite morphisms are both projective and affine.[9]
Morphisms of finite type[]
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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-module 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 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[]
- Glossary of algebraic geometry
- Finite algebra
Notes[]
- ^ Shafarevich 2013, p. 60, Def. 1.1.
- ^ Shafarevich 2013, p. 62, Def. 1.2.
- ^ 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
- Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer Science. ISBN 978-0-387-97716-4.
External links[]
- The Stacks Project Authors, The Stacks Project
- Algebraic geometry
- Morphisms