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To prove this we switch to the ideal version of this fact. We look at the affine case and
leave the projective case as an exercise. Specifically, it suffices to show that if I and J
are prime ideals in k[
V
] of transcendence degree e and f over k, respectively, with
I … J, then we can find a chain of distinct prime ideals
IJ
………
...
J
=
J
1
ef
-
of length e - f. We sketch the proof in [Kend77].
We may assume that
[
]
kX X
,
,...,
X
12
n
[]
=
[
]
k
=
kx x
,
,...,
x
,
12
n
()
IV
[
]
Ikyy
=
,
,...,
y
,
12
n
[
]
Jkzz
=
,
,...,
z
,
12
n
where x
i
is the projection of X
i
in k[
V
], and that y
1
, y
2
,..., y
e
and z
1
= y
1
, z
2
= y
2
,...,
z
f
= y
f
are a transcendence basis of I and J over k, respectively. Furthermore, we may
assume that the natural projection homomorphism
I
IVJ
p :I
Æ
=
J
(
()
)
sends y
i
to z
i
under the appropriate identifications.
Because the y
i
are a transcendental basis for k[y
1
,y
2
,...,y
e
] over k, there is a
unique ring homomorphism (over k)
[
]
Æ
s :
ky y
,
,...,
y
I
.
12
e
defined by
()
=
s yz
,
,
1
£
i
£
f
+
1
i
i
=
yf
+
1
<
i
£
e
.
i
Claim.
The homomorphism s extends to a unique ring homomorphism
[
]
Æ
s
1
:
Ikyy
=
,
,...,
y
I
.
1
2
n
To prove the claim, note that y
s
is algebraic over k[y
1
,y
2
,...,y
e
] for e < s £ n. Let
(
)
Œ
[
][
]
pyy
,
,...,
y X kyy
,
,
,...,
y X
s
12
e
12
e
be its minimal polynomial. Applying the projection map p, we see that p
s
(z
1
,z
2
,
...,z
e
,X) has positive degree, which in turn implies that p
s
(z
1
,z
2
,...,z
f+1
,y
f+2
,...,y
e
,X)
has positive degree. The claim now follows from Theorem B.8.15.
Let J
1
=s
1
(I). The ideal J
1
is prime. Since z
t
is algebraic over k[z
1
,z
2
,...,z
f
] for
t > f, it follows that J
1
has transcendence degree e - 1 over k. Repeating the same steps
for J
1
, produces a prime ideal J
2
, J
1
… J
2
, of transcendence degree e - 2 over k, and so
on until we finally get J
e-f
= J.
Theorem 10.16.9 is proved.
Let us expand on the definition of dimension of a projective variety
V
Õ
P
n
(
C
) in
terms of the existence of a finite map
