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''
1, ..., ''xn''">'x''1, ..., ''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 ''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 In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used ...
over ''k'' is of finite type over ''k''.
The
Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negati ...
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
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
''X'' over a field has a finite surjective morphism to
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P
''n'', where ''n'' is the dimension of ''X''.
See also
*
Finitely generated algebra In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element of ...
References
{{Cite book , last=Bosch , first=Siegfried , title=Algebraic Geometry and Commutative Algebra , publisher=
Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
, year=2013 , isbn=9781447148289 , location=London , pages=360–365
Algebraic geometry
Morphisms