finite morphism#morphisms of finite type

A morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes

:$V\_i\; =\; \backslash mbox\{Spec\}\; \backslash ;\; B\_i$

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

:$B\_i\; \backslash rightarrow\; A\_i,$

makes A_i a finitely generated module over B_i (in other words, a finite B_i-algebra).{{sfn|Hartshorne|1977|loc=Section II.3}} One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine 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.{{Citation | title=Stacks Project, Tag 01WG | url=http://stacks.math.columbia.edu/tag/01WG}}.

For example, for any field k, $\backslash text\{Spec\}(k[t,x]/(x^n-t))\; \backslash to\; \backslash text\{Spec\}(k[t])$ is a finite morphism since $k[t,x]/(x^n-t)\; \backslash cong\; k[t]\backslash oplus\; k[t]\backslash cdot\; x\; \backslash oplus\backslash cdots\; \backslash oplus\; k[t]\backslash cdot\; x^\{n-1\}$ 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.

• 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 (section of the ideal sheaf) 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.Grothendieck, EGA IV, Part 4, Corollaire 18.12.4. 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.Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
• Finite morphisms are both projective and affine.

• Glossary of algebraic geometry
• Morphism of finite type

{{reflist|1=30em}}

{{refbegin}}

• {{EGA|book=4-3| pages = 5–255}}
• {{EGA|book=4-4| pages = 5–361}}
• {{Hartshorne AG}}
• {{cite book | last=Shafarevich | first=Igor R. | author-link=Igor Shafarevich| title = Basic Algebraic Geometry 1 | year=2013| publisher=Springer Science | doi=10.1007/978-3-642-37956-7 | url=https://link.springer.com/book/10.1007/978-3-642-37956-7 | isbn=978-0-387-97716-4}}

{{refend}}

Category:Algebraic geometry

Category:Morphisms