16
LECTURE 3.
MORPHISMS OF AFFINE ALGEBRAIC VARIETIES
Note that it suffices to check the previous condition only for generators of the
ideal
I
(
Y
)
, for example for the polynomials defining the system of equations
Y
.
In terms of the polynomials
(
P
1
(
T
)
, . . . , P
m
(
T
))
satisfying (
3.5
), the morphism
f
:
X
→
Y
is given as follows:
f
K
(
a
) = (
P
1
(
a
)
, . . . , P
m
(
a
))
∈
Y
(
K
)
,
∀
a
∈
X
(
K
)
.
It follows from the definitions that a morphism
φ
given by polynomials
((
P
1
(
T
)
, . . . , P
m
(
T
))
satisfying (
3.5
) is an isomorphism if and only if there exist
polynomials
(
Q
1
(
T
)
, . . . , Q
n
(
T
))
such that
G
(
Q
1
(
T
)
, . . . , Q
n
(
T
))
∈
I
(
Y
)
,
∀
G
∈
I
(
X
)
,
P
i
(
Q
1
(
T
)
, . . . , Q
n
(
T
))
≡
T
i
mod
I
(
Y
)
, i
= 1
, . . . , m,
Q
j
(
P
1
(
T
)
, . . . , P
m
(
T
))
≡
T
j
mod
I
(
X
)
, j
= 1
, . . . , n.
The main problem of (affine) algebraic geometry is to classify affine algebraic
varieties up to isomorphism. Of course, this is a hopelessly difficult problem.
Example 3.3.
1. Let
Y
be given by the equation
T
2
1

T
3
2
= 0
,
and
X
=
A
1
k
with
O
(
X
) =
k
[
T
]
. A morphism
f
:
X
→
Y
is given by the pair of polynomials
(
T
3
, T
2
)
. For every
k
algebra
K
,
f
K
(
a
) = (
a
3
, a
2
)
∈
Y
(
K
)
, a
∈
X
(
K
) =
K.
The affine algebraic varieties
X
and
Y
are not isomorphic since their coor
dinate rings are not isomorphic.
The quotient field of the algebra
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 Spring '09
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 Algebra, Geometry, Polynomials, Equations, Category theory, aﬃne algebraic varieties

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