Morphism of finite type explained
of
commutative rings,
is called an
-
algebra of
finite type if
is a
finitely generated as an
-algebra. It is much stronger for
to be a
finite
-algebra, which means that
is finitely generated as an
-
module. For example, for any commutative ring
and natural number
, the
polynomial ring
is an
-algebra of finite type, but it is not a finite
-module unless
= 0 or
= 0. Another example of a finite-type homomorphism that is not finite is
C[t]\toC[t][x,y]/(y2-x3-t)
.
of schemes is of
finite type if
has a covering by
affine open subschemes
Vi=\operatorname{Spec}(Ai)
such that
has a finite covering by affine open subschemes
Uij=\operatorname{Spec}(Bij)
of
with
an
-algebra of finite type. One also says that
is of
finite type over
.
For example, for any natural number
and
field
,
affine
-space and
projective
-space over
are of finite type over
(that is, over
), while they are not finite over
unless
= 0. More generally, any
quasi-projective scheme over
is of finite type over
.
The Noether normalization lemma says, in geometric terms, that every affine scheme
of finite type over a field
has a finite surjective morphism to affine space
over
, where
is the
dimension of
. Likewise, every projective scheme
over a field has a finite surjective morphism to
projective space
, where
is the dimension of
.
See also
References
Book: Bosch, Siegfried . Siegfried Bosch
. Siegfried Bosch. Algebraic Geometry and Commutative Algebra . . 2013 . 9781447148289 . London . 360–365.
