Graphics Reference
In-Depth Information
p
q
p
q
Ê
Ë
ˆ
¯
1
1
m
m
f
,...,
.
The map
[]
Æ
()
k
k
uf u
WV
Æ
o
is a ring homomorphism over k that extends to a unique homomorphism
()
Æ
()
uk
*:
WV
k
.
Definition.
The map u* is called the
pullback map
defined by u.
10.13.12. Proposition.
The pullback map u* is a well-defined homomorphism
over k.
Proof.
See [Shaf94].
10.13.13. Proposition.
Let u :
V
Æ
W
and v :
W
Æ
U
be rational functions between
irreducible affine varieties
V
,
W
, and
U
.
(1) If u is dominant, then u* is one-to-one.
(2) If both u and v are dominant, then
vu
o
:
VU
Æ
is a dominant rational function and
(
)
=** .
vu
o
u v
o
Proof.
See [Shaf94].
Definition.
A dominant rational map j :
V
Æ
W
between irreducible affine varieties
V
and
W
is said to be
birational
if j has a dominant rational inverse, that is, there is
a dominant rational map y :
W
Æ
V
, so that j
°
y and y
°
j are the identity maps wher-
ever they are defined. We call
V
and
W
birationally equivalent
if there is a birational
map j :
V
Æ
W
. An affine variety is called
rational
if it is birationally equivalent to k
n
for some n.
10.13.14. Theorem.
Two irreducible affine varieties are birationally equivalent if
and only if they have isomorphic rational function fields.
Proof.
See [Shaf94] or [CoLO97].
10.13.15. Theorem.
Every irreducible affine variety is birationally equivalent to a
hypersurface in some k
n
.
