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87
Example 10.4.
1. Let
X
=
{
(
x, y
)
∈
K
2
:
y
=
x
2
} ⊂
A
2
(
K
)
and
Y
=
A
1
(
K
)
.
Consider the projection map
f
:
X
→
Y,
(
x, y
)
7→
y
. Then
f
is ﬁnite. Indeed,
O
(
X
)
∼
=
k
[
Z
1
, Z
2
]
/
(
Z
2

Z
2
1
)
,
O
(
Y
)
∼
=
k
[
Z
2
]
and
f
*
is the composition of the natural inclusion
k
[
Z
2
]
→
k
[
Z
1
, Z
2
]
and the natural homomorphism
k
[
Z
1
, Z
2
]
→
k
[
Z
1
, Z
2
]
/
(
Z
2

Z
2
1
)
.
Obviously, it is injective. Let
z
1
, z
2
be the images of
Z
1
and
Z
2
in the factor ring
k
[
Z
1
, Z
2
]
/
(
Z
2

Z
2
1
)
. Then
O
(
X
)
is generated over
f
*
(
O
(
Y
))
by one element
z
1
.
The latter satisﬁes a monic equation:
z
2
1

f
*
(
Z
2
) = 0
with coeﬃcients in
f
*
(
O
(
Y
))
.
As we saw in the proof of Lemma
10.1
, this implies that
O
(
X
)
is a ﬁnitely generated
f
*
(
O
(
Y
))
module and hence
O
(
X
)
is integral over
f
*
(
O
(
Y
))
. Therefore
f
is a ﬁnite
map.
2. Let
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This note was uploaded on 01/08/2012 for the course MATH 299 taught by Professor Wei during the Spring '09 term at SUNY Stony Brook.
 Spring '09
 wei
 Algebra, Geometry

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