Cryptography Reference
In-Depth Information
V
(
w
2
x
2
w
2
z
2
y
2
z
2
x
2
z
2
)
3
(
(
x
2
Exercise
5.4.7
Let
X
=
−
−
+
⊆ P
k
).
Show
that
+
yz
)
/
(
x
2
w
2
)
(
x
2
z
2
)
/
(
x
2
(
X
). Hence, find the value of (
x
2
z
2
)
/
(
x
2
−
≡
−
−
yz
)in
k
−
−
yz
) at the point (
w
:
x
:
y
:
z
)
(0 : 1 : 1 : 1). Show that both representations of the func-
tion have the same value on the point (
w
:
x
:
y
:
z
)
=
=
(2 : 1 :
−
1:1).
Theorem 5.4.8
Let X be a variety and let f be a rational function. Then there is a non-
empty open set U
⊂
X such that f is regular on U. Conversely, if U
⊂
X is non-empty and
open and f
:
U
is a function given by a ratio f
1
/f
2
of polynomials (homogeneous
polynomials of the same degree if X is projective) that is defined for all P
→ k
∈
U then f
extends uniquely to a rational function f
:
X
→ k
.
Proof
Let
f
=
f
1
/f
2
where
f
1
,f
2
∈ k
[
X
]. Define
U
=
X
−
V
(
f
2
). Since
f
2
=
0in
k
[
X
]
we have
U
a non-empty open set, and
f
is regular on
U
.
For the converse let
f
f
1
/f
2
be a function on
U
given as a ratio of polynomials. Then
one can consider
f
1
and
f
2
as elements of
=
k
[
X
] and
f
2
non-zero on
U
implies
f
2
=
0
in
(
X
). Finally, suppose
f
1
/f
2
and
f
3
/f
4
are functions on
X
(where
f
1
,f
2
,f
3
,f
4
are polynomials) such that the restrictions
(
f
1
/f
2
)
k
[
X
]. Hence,
f
1
/f
2
corresponds to an element of
k
|
U
and (
f
3
/f
4
)
|
U
are equal. Then
f
1
f
4
−
f
2
f
3
is zero on
U
and, by Lemma
5.3.10
,
(
f
1
f
4
−
f
2
f
3
)
∈
I
k
(
X
) and
f
1
/f
2
≡
f
3
/f
4
on
X
.
(
X
)
= k
n
n
)
.If
Corollary 5.4.9
If X is a projective variety
an
d X
∩ A
= ∅
then
k
(
X
∩ A
(
X
)
=
k
k
X is non-empty affine variety then
(
X
)
.
∩ A
n
=
−
Proof
The result follows since
X
X
V
(
x
n
) is open in
X
and
X
is open in
X
.
⊆
O
k
Definition 5.4.10
Let
X
be a
v
ariety and
U
X
. Define
(
U
) to be the elements of
(
X
)
that are regular on all
P
∈
U
(
k
).
Lemma 5.4.11
If X is an affine variety over
k
then
O
(
X
)
= k
[
X
]
.
Proof
Theorem I.3.2 of Hartshorne [
252
].
Definition 5.4.
12
Let
X
be a variety over
k
and
f
∈ k
(
X
). Let
σ
∈
Gal(
k
/
k
). If
f
=
f
1
/f
2
where
f
1
,f
2
∈ k
[
x
] define
σ
(
f
)
=
σ
(
f
1
)
/σ
(
f
2
) where
σ
(
f
1
) and
σ
(
f
2
) denote the natural
Galois action on polynomials (i.e.,
σ
(
i
a
i
x
i
)
=
i
σ
(
a
i
)
x
i
). Some authors write this
as
f
σ
.
f
then
σ
(
f
)
σ
(
f
)). Let
Exercis
e
5.4.13
Prove that
σ
(
f
) is well-defined (i.e., if
f
≡
≡
P
∈
X
(
k
). Prove that
f
(
P
)
=
0 if and only if
σ
(
f
)(
σ
(
P
))
=
0.
Remark
5
.4.14
Having
d
efined an action of
G
=
Ga
l(
k
/
k
)on
k
(
X
) it is natural to ask
(
X
)
G
whether
k
={
f
∈ k
(
X
):
σ
(
f
)
=
f
∀
σ
∈
Gal(
k
/
k
)
}
is the same as
k
(
X
). The issue
is whether a function being “defined over
k
” is the same as “can be written with coefficients
in
k
”. Indeed, this is true, but not completely trivial.
