Scheme 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''-module unless ''A'' = 0 or ''n'' = 0. Another example of a finite-type morphism which is not finite is \mathbb \to \mathbb 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 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

* Finitely generated algebra

References

{{Cite book , last=Bosch , first=Siegfried , title=Algebraic Geometry and Commutative Algebra , publisher=Springer , year=2013 , isbn=9781447148289 , location=London , pages=360–365

Algebraic geometry
Morphisms