Cryptography Reference
In-Depth Information
A sketch of the argument is given in Exercise 1.12 of Silverman [
505
] and we give a
few extra hints here. Let
X
be a projective variety. One first shows that if
X
is defined
over
k
is a finite Galois extension of
k
and if
k
then
I
k
(
X
) is an induced Galois module
k
/
(see page 110 of Serre [
488
]) for Gal(
k
). It follows from Section VII.1 of [
488
] that
the Galois cohomology group
H
1
(Gal(
k
/
k
)
,I
k
(
X
)) is trivial and hence, by Section X.3 of
[
488
],
that
H
1
(
G,I
k
(
X
))
=
0. One can therefore deduce, as in Exercise 1.12(a) of [
505
],
[
X
]
G
that
k
= k
[
X
].
(
X
)
G
1
and let
σ
To show that
k
= k
(
X
)let(
f
0
:
f
1
):
X
→ P
∈
G
. Then
σ
(
f
0
)
=
λ
σ
f
0
+
∗
and
G
0
,σ
,G
1
,σ
∈
G
0
,σ
and
σ
(
f
1
)
=
λ
σ
f
1
+
G
1
,σ
where
λ
σ
∈ k
I
k
(
X
). One shows fir
s
t
∗
), which is trivial by Hilbert 90, and so
λ
σ
=
that
λ
σ
∈
H
1
(
G,
k
∈ k
σ
(
α
)
/α
for some
α
.
Replacing
f
0
by
αf
0
and
f
1
by
αf
1
gives
λ
σ
=
1 and one can proceed to showing that
G
0
,σ
,G
1
,σ
∈
0 as above. The result follows.
For a different approach see Theorem
7.8.3
and Remark
8.4.8
below, or Corollary 2,
Section VI.5 (page 178) of Lang [
326
].
H
1
(
G,I
k
(
X
))
=
5.5 Rational maps and morphisms
Definition 5.5.1
Let
X
be an affine or projective variety over a field
k
and
Y
an affine
n
over
n
of the form
variety in
A
k
.Let
φ
1
,...,φ
n
∈ k
(
X
). A map
φ
:
X
→ A
=
φ
(
P
)
(
φ
1
(
P
)
,...,φ
n
(
P
))
(5.1)
is
regular
at a point
P
∈
X
(
k
)ifall
φ
i
,for1
≤
i
≤
n
, are regular at
P
.A
r
at
ional map
φ
:
X
→
Y
defined over
k
is
a
map of the form (
5.1
) such that, for all
P
∈
X
(
k
)forwhich
φ
is regular at
P
,
φ
(
P
)
).
Let
X
be an affine or projective variety over a field
∈
Y
(
k
n
k
and
Y
a projective variety in
P
n
of the form
over
k
.Let
φ
0
,...,φ
n
∈ k
(
X
). A map
φ
:
X
→ P
φ
(
P
)
=
(
φ
0
(
P
):
···
:
φ
n
(
P
))
(5.2)
∈
k
∈ k
is
regular
at a point
P
X
(
) if there is some function
g
(
X
) such that all
gφ
i
,for
0
≤
i
≤
n
, are regular at
P
and, for some 0
≤
i
≤
n
, one has (
gφ
i
)(
P
)
=
0.
6
A
ra
ti
onal
map
φ
:
X
→
Y
defined over
k
is
a
map of the form (
5.2
) such that, for all
P
∈
X
(
k
)for
which
φ
is regular at
P
,
φ
(
P
)
∈
Y
(
k
).
We stress that a rational map is not necessarily defined at every point of the domain. In
other words, it is not necessarily a function.
Exercise 5.5.2
Let
X
and
Y
be projective varieties. Show that one can write a rational
map in the form
φ
(
P
)
=
(
φ
0
(
P
):
···
:
φ
n
(
P
)) where the
φ
i
(
x
)
∈ k
[
x
] are all homogeneous
polynomials of the same degree, not all
φ
i
(
x
)
∈
I
k
(
X
), and for every
f
∈
I
k
(
Y
)wehave
f
(
φ
0
(
x
)
,...,φ
n
(
x
))
∈
I
k
(
X
).
6
n
.
This last condition is to prevent
φ
mapping to (0 :
···
: 0), which is not a point in
P
