Graphics Reference
In-Depth Information
y
i
:
W
Æ
k
.
It follows that h(y
i
) will be represented by some polynomial a
i
(X
1
,X
2
,...,X
n
) Πk[X
1
,
X
2
,..., X
n
]. One needs to show that the map
u:
VW
Æ
defined by
(
)
=
(
(
)
(
)
(
)
)
u x
,
x
,...,
x
a
x
,
x
,...,
x
,
a
x
,
x
,...,
x
,...,
a
x
,
x
,...,
x
12
n
112
n
212
n
m
12
n
is the unique map that does the job. For more details see [CoLO97].
A fact that will be useful later is
10.13.4. Proposition.
Let f :
V
Æ
W
be a polynomial function between affine vari-
eties. Then f(
V
) is dense in
W
with respect to the Zariski topology if and only if
f* : k[
W
] Æ k[
V
] is one-to-one.
Proof.
See [CoLO97]. The “only if” part is easy.
Let
V
Õ k
n
and
W
Õ k
m
be affine varieties. A polynomial function
Definition.
u:
VW
Æ
is called an
isomorphism
if u has an inverse that is also a polynomial function. In that
case the two varieties are said to be
isomorphic
.
10.13.5. Theorem.
Two affine varieties
V
Õ k
n
and
W
Õ k
m
are isomorphic if and
only their coordinate rings k[
V
] and k[
W
] are isomorphic over k.
Proof.
Only the “if” part is nontrivial. Given an isomorphism of coordinate rings we
must produce a polynomial function from
V
to
W
which is an isomorphism. Theorem
10.13.3 does that.
10.13.6. Example.
To show that the graph of a polynomial function f(X,Y), namely
V(Z - f(X,Y)) Ã k
3
, is isomorphic to k
2
.
Solution.
Consider the bijections
p
=-
(
(
)
)
()
V
VZ fXY
,
W
=
VZ
,
s
where
(
)
=
(
)
p
s
xyz
,,
xy
,,
0
(
)
=
(
(
)
)
xy
,,
0
xyfxy
,,
, .
