Cryptography Reference
In-Depth Information
3. V
(
f
)
∩
V
(
g
)
=
V
(
f,g
)
.
n
(
4. If X
1
⊆
X
2
⊆ P
k
)
then I
k
(
X
2
)
⊆
I
k
(
X
1
)
⊆ k
[
x
0
,...,x
n
]
.
5. I
k
(
X
1
∪
I
k
(
X
2
)
.
6. If J is a homogeneous ideal then J
X
2
)
=
I
k
(
X
1
)
∩
I
k
(
V
(
J
))
.
7. If X is a projective algebraic set defined over
⊆
k
then V
(
I
k
(
X
))
=
X.IfY is another
projective algebraic set defined over
k
and I
k
(
Y
)
=
I
k
(
X
)
then Y
=
X.
Exercise 5.2.13
Prove Proposition
5.2.12
.
Definition 5.2.14
If
X
is a projective algebraic set defined over
k
then the
homogeneous
coordinate ring
of
X
over
k
is
k
[
X
]
= k
[
x
0
,...,x
n
]
/I
k
(
X
).
Note that elements of
k
[
X
] are not necessarily homogeneous polynomials.
n
(respectively,
n
)The
Zariski topology
Definition 5.2.15
Let
X
be an algebraic set in
A
P
is the topology on
X
defined as follows: the closed sets are
X
∩
Y
for every algebraic set
n
(respectively,
Y
n
).
Y
⊆ A
⊆ P
Exercise 5.2.16
Show that the Zariski topology satisfies the axioms of a topology.
n
:
x
i
=
n
Definition 5.2.17
For 0
≤
i
≤
n
define
U
i
={
(
x
0
:
···
:
x
n
)
∈ P
0
}=P
−
V
(
x
i
).
(These are open sets in the Zariski topology.)
n
n
Exercise 5.2.18
Show that
i
=
0
U
i
(not a disjoint union).
We already mentioned the map
ϕ
:
P
=∪
n
n
given by
ϕ
(
x
1
,...,x
n
)
:
x
n
:1),
which has image equal to
U
n
. A useful way to study a projective algebraic set
X
is to consider
X
A
→ P
=
(
x
1
:
···
∩
U
i
for 0
≤
i
≤
n
and interpret
X
∩
U
i
as an affine algebraic set. We now introduce the
notation for this.
n
(
Definition 5.2.19
Let
ϕ
i
:
A
k
)
→
U
i
be the one-to-one correspondence
ϕ
i
(
y
1
,...,y
n
)
=
(
y
1
:
···
:
y
i
:1:
y
i
+
1
:
···
:
y
n
)
.
We write
ϕ
for
ϕ
n
.Let
ϕ
−
1
i
(
x
0
:
···
:
x
n
)
=
(
x
0
/x
i
,...,x
i
−
1
/x
i
,x
i
+
1
/x
i
,...,x
n
/x
i
)
be the map
ϕ
−
1
i
), which is defined only on
U
i
(i.e.,
ϕ
−
1
ϕ
−
1
i
:
P
n
(
k
)
→ A
n
(
k
(
X
)
=
(
X
∩
i
U
i
)).
3
We write
X
n
as an abbreviation for
ϕ
−
n
(
X
∩ A
∩
U
n
).
Let
ϕ
i
:
[
y
1
,...,y
n
]bethe
de-homogenisation
map
4
k
[
x
0
,...,x
n
]
→ k
ϕ
i
(
f
)(
y
1
,...,y
n
)
=
f
◦
ϕ
i
(
y
1
,...,y
n
)
=
f
(
y
1
,...,y
i
,
1
,y
i
+
1
,...,y
n
)
.
We write
ϕ
∗
for
ϕ
n
.
This notation does not seem to be standard. Our notation agrees with Silverman [
505
], but Hartshorne [
252
]has
ϕ
i
and
ϕ
−
i
the
other way around.
3
4
The upper star notation is extended in Definition
5.5.20
.
