In commutative algebra, given a homomorphism
A
→
B
{\displaystyle A\to B}
of commutative rings,
B
{\displaystyle B}
is called an
A
{\displaystyle A}
-algebra of finite type if
B
{\displaystyle B}
can be finitely generated as an
A
{\displaystyle A}
-algebra. It is much stronger for
B
{\displaystyle B}
to be a finite
A
{\displaystyle A}
-algebra, which means that
B
{\displaystyle B}
is finitely generated as an
A
{\displaystyle A}
-module. For example, for any commutative ring
A
{\displaystyle A}
and natural number
n
{\displaystyle n}
, the polynomial ring
A
[
x
1
,
…
,
x
n
]
{\displaystyle A[x_{1},\dots ,x_{n}]}
![{\displaystyle A[x_{1},\dots ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2e6fd42bca14e9063ee5a2ab5da6e2599e9639)
is an
A
{\displaystyle A}
-algebra of finite type, but it is not a finite
A
{\displaystyle A}
-algebra unless
A
{\displaystyle A}
= 0 or
n
{\displaystyle n}
= 0. Another example of a finite-type homomorphism that is not finite is
C
[
t
]
→
C
[
t
]
[
x
,
y
]
/
(
y
2
−
x
3
−
t
)
{\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)}
![{\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9baac4fec1c0ac773861cf0b6abf928b4f8a79a8)
.
The analogous notion in terms of schemes is that a morphism
f
:
X
→
Y
{\displaystyle f:X\to Y}
of schemes is of finite type if
Y
{\displaystyle Y}
has a covering by affine open subschemes
V
i
=
Spec
(
A
i
)
{\displaystyle V_{i}=\operatorname {Spec} (A_{i})}
such that
f
−
1
(
V
i
)
{\displaystyle f^{-1}(V_{i})}
has a finite covering by affine open subschemes
U
i
j
=
Spec
(
B
i
j
)
{\displaystyle U_{ij}=\operatorname {Spec} (B_{ij})}
of
X
{\displaystyle X}
with
B
i
j
{\displaystyle B_{ij}}
an
A
i
{\displaystyle A_{i}}
-algebra of finite type. One also says that
X
{\displaystyle X}
is of finite type over
Y
{\displaystyle Y}
.
For example, for any natural number
n
{\displaystyle n}
and field
k
{\displaystyle k}
, affine
n
{\displaystyle n}
-space and projective
n
{\displaystyle n}
-space over
k
{\displaystyle k}
are of finite type over
k
{\displaystyle k}
(that is, over
Spec
(
k
)
{\displaystyle \operatorname {Spec} (k)}
), while they are not finite over
k
{\displaystyle k}
unless
n
{\displaystyle n}
= 0. More generally, any quasi-projective scheme over
k
{\displaystyle k}
is of finite type over
k
{\displaystyle k}
.
The Noether normalization lemma says, in geometric terms, that every affine scheme
X
{\displaystyle X}
of finite type over a field
k
{\displaystyle k}
has a finite surjective morphism to affine space
A
n
{\displaystyle \mathbf {A} ^{n}}
over
k
{\displaystyle k}
, where
n
{\displaystyle n}
is the dimension of
X
{\displaystyle X}
. Likewise, every projective scheme
X
{\displaystyle X}
over a field has a finite surjective morphism to projective space
P
n
{\displaystyle \mathbf {P} ^{n}}
, where
n
{\displaystyle n}
is the dimension of
X
{\displaystyle X}
.