finite morphism

In algebraic geometry, a finite morphism between two affine varieties $X, Y$ is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is integral over k\left \right/math>. This definition can be extended to the quasi-projective varieties, such that a regular map $f\colon X\to Y$ between quasiprojective varieties is finite if any point $y\in Y$ has an affine neighbourhood V such that $U=f^(V)$ is affine and $f\colon U\to V$ is a finite map (in view of the previous definition, because it is between affine varieties).

Definition by schemes

A morphism ''f'': ''X'' → ''Y'' of schemes is a finite morphism if ''Y'' has an open cover by affine schemes
:$V_i = \mbox \; B_i$

such that for each ''i'',

:$f^(V_i) = U_i$

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'' (in other words, a finite ''B''_''i''-algebra). 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..

For example, for any field ''k'', \text(k ,x(x^n-t)) \to \text(k is a finite morphism since k ,x(x^n-t) \cong k oplus k cdot x \oplus\cdots \oplus k cdot x^ as k /math>-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 ''A''¹ − 0 into ''A''¹ is not finite. (Indeed, the Laurent polynomial ring ''k'' 'y'', ''y''⁻¹is 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 (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. 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.

See also

* Glossary of algebraic geometry
*Morphism of finite type

Notes

References

*
*
*
*
{{refend

Algebraic geometry
Morphisms