Cryptography Reference
In-Depth Information
5.6 Dimension
The natural notion of dimension (a point has dimension 0, a line has dimension 1, a plane
has dimension 2, etc.) generalises to algebraic varieties. There are algebraic and topological
ways to define dimension. We use an algebraic approach.
8
We stress that the notion of dimension only applies to irreducible algebraic sets. For
example,
X
2
=
V
(
x,y
)
∪
V
(
x
−
1)
=
V
(
x
(
x
−
1)
,y
(
x
−
1))
⊆ A
is the union of a point
and a line so has components of different dimension.
Recall the notion of transcendence degree of an extension
k
(
X
) over
k
from Defini-
tion
A.6.3
.
Definition 5.6.1
Let
X
be a variety over
k
.The
dimension
of
X
, denoted dim(
X
), is the
transcendence degree of
k
(
X
) over
k
.
Example 5.6.2
The dimension of
A
n
is
n
. The dimension of
P
n
is
n
.
Theorem 5.6.3
Let X and Y be varieties. If X and Y are birationally equivalent then
dim(
X
)
=
dim(
Y
)
.
Proof
Immediate from Theorem
5.5.25
.
n
is non-empty. Then
Corollary 5.6.4
Let X be a projective variety such that X
∩ A
dim
X
.
n
)
. Let X be an affine variety. Then
dim(
X
)
dim(
X
)
=
dim(
X
∩ A
=
n
.
Exercise 5.6.5
Let
f
be a non-constant polynomial and let
X
=
V
(
f
) be a variety in
A
Show that dim(
X
)
=
n
−
1.
Exercise 5.6.6
Show that if
X
is a non-empty variety of dimension 0 then
X
={
P
}
is a
single point.
A useful alternative formulation of dimension is as follows.
Definition 5.6.7
Let
R
be a ring. The
Krull dimension
of
R
is the supremum of
n
∈ Z
≥
0
such that there exists a chain
I
0
⊂
I
1
⊂···⊂
I
n
of prime
R
-ideals such that
I
j
−
1
=
I
j
for
1
≤
j
≤
n
.
k
Theorem 5.6.8
Let X be an affine variety over
. Then
dim(
X
)
is equal to the Krull
k
dimension of the affine coordinate ring
[
X
]
.
Proof
See Proposition I.1.7 and Theorem I.1.8A of [
252
].
Corollary 5.6.9
Let X and Y be affine varieties over
k
such that Y is a proper subset of
X. Then
dim(
Y
)
<
dim(
X
)
.
Proof
Since
Y
I
k
(
Y
) and both ideals are prime since
X
and
Y
are
irreducible. It follows that the Krull dimension of
=
X
we have
I
k
(
X
)
k
[
X
] is at least one more than the Krull
dimension of
k
[
Y
].
Exercise 5.6.10
Show that a proper closed subset of a variety of dimension 1 is finite.
8
See Chapter 8 of Eisenbud [
176
] for a clear criticism of this approach.
