Cryptography Reference
In-Depth Information
Lemma 5.5.19
LetXandY be affine varieties over
k
,letU
⊆
Xbe open and letφ
:
U
→
Y
be a morphism. Then φ is dominant if and only if φ
∗
is injective.
then
φ
∗
:
(
U
) restricts to
φ
∗
:
Note that if
X
and
φ
are defined over
k
k
[
Y
]
→
O
k
[
Y
]
→
(
X
). If
φ
∗
is injective then one can extend it to get a homomorphism of the field of fractions
of
k
k
[
Y
]to
k
(
X
).
k
→
Definition 5.5.20
Let
X
and
Y
be varieties over
and let
φ
:
X
Y
be a dominant rational
. Define the
pullback
φ
∗
:
(
X
)by
φ
∗
(
f
)
map defined over
k
k
(
Y
)
→ k
=
f
◦
φ
.
We will now state that
φ
∗
is a
k
-algebra homomorphism. Recall that a
k
-algebra homo-
morphism of fields is a field homomorphism that is the identity map on
k
.
Theorem 5.5.21
Let X and Y be varieties over
k
and let φ
:
X
→
Y be a dominant
. Then the pullback φ
∗
:
rational map defined over
k
k
(
Y
)
→ k
(
X
)
is an injective
k
-algebra
homomorphism.
Proof
Proposition 6.11 of Fulton [
199
] or Theorem I.4.4 of Hartshorne [
252
].
Example 5.5.22
Consider the rational maps from Example
5.5.16
.Themap
φ
(
x,y
)
=
k
k
(
x,x
) is not dominant and does not induce a well-defined function from
(
x,y
)to
(
x,y
)
since, for example,
φ
∗
(1
/
(
x
−
=
−
=
y
))
1
/
(
x
x
)
1
/
0.
The map
φ
(
x,y
)
=
(
x,xy
) is dominant and
φ
∗
(
f
(
x,y
))
=
f
(
x,xy
) is a field isomor-
phism.
Exercise 5.5.23
Let
K
1
,K
2
be fields containing a field
k
.Let
θ
:
K
1
→
K
2
be a
k
-algebra
homomorphism. Show that
θ
is injective.
Theorem 5.5.24
Let X and Y be varieties over
k
and let θ
:
k
(
Y
)
→ k
(
X
)
be a
k
-algebra
homomorphism. Then θ induces a dominant rational map φ
:
X
→
Y defined over
k
.
Proof
Proposition 6.11 of Fulton [
199
] or Theorem I.4.4 of Hartshorne [
252
].
Theorem 5.5.25
Let X andY be varieties over
k
. ThenX andY are birationally equivalent
(
X
)
=
k
k
k
over
if and only if
(
Y
)
(isomorphic as fields).
Proof
Proposition 6.12 of Fulton [
199
] or Corollary I.4.5 of Hartshorne [
252
].
Some authors prefer to study function fields rather than varieties, especially in the case
of dimension 1 (there are notable classical texts that take this point of view by Chevalley
and Deuring; see Stichtenoth [
529
] for a more recent version). By Theorem
5.5.25
(and
other results), the study of function fields up to isomorphism is the study of varieties up to
birational equivalence. A specific set of equations to describe a variety is called a
model
.
