Cryptography Reference
In-Depth Information
3.
m
P,
k
(
X
)
is a maximal ideal;
4.
O
P,
k
(
X
)
is a Noetherian local ring.
Proof
The first three parts are straightforward. The fourth part follows from the fact that, if
X
is affine,
O
P,
k
(
X
) is the localisation of
k
[
X
] (which is Noetherian) at the maximal ideal
m
={
. Lemma
A.9.5
shows that the localisation of a Noetherian ring
at a maximal ideal is Noetherian. Similarly, if
X
is projective then
f
∈ k
[
X
]:
f
(
P
)
=
0
}
O
P,
k
(
X
) is isomorphic
[
ϕ
−
1
i
to a localisation of
R
= k
(
X
)] (again, Noetherian) where
i
is such that
P
∈
U
i
.
Note that, for an affine variety
X
k ⊆ k
[
X
]
⊆
O
P
(
X
)
⊆ k
(
X
)
.
Remark 7.1.3
We remark that
O
P,
k
(
X
) and m
P,
k
(
X
) are defined in terms of
k
(
X
) rather
than any particular model for
X
. Hence, if
φ
:
X
→
Y
is a birational map over
k
of
varieties over
k
and
φ
is defined at
P
∈
X
(
k
) then
O
P,
k
(
X
) is isomorphic as a ring
∈
O
φ
(
P
)
,
k
(
Y
) then
φ
∗
(
f
)
to
O
φ
(
P
)
,
k
(
Y
) (precisely, if
f
=
f
◦
φ
∈
O
P,
k
(
X
)). Similarly,
m
P,
k
(
X
) and m
φ
(
P
)
,
k
(
Y
) are isomorphic.
Let
X
be a projective variety, let
P
∈
X
(
k
), and let
i
such that
P
∈
U
i
. By Corollary
5.4.9
,
(
X
)
=
k
O
P,
k
(
X
)
=
O
ϕ
−
i
(
P
)
,
k
(
ϕ
−
1
i
(
ϕ
−
1
i
k
U
i
)). It is therefore sufficient to
consider affine varieties when determining local properties of a variety.
(
X
)) and so
(
X
∩
n
be an affine variety and suppose
P
Example 7.1.4
Let
X
⊆ A
=
(0
,...,
0)
∈
X
(
k
). Then
O
P
=
O
P,
k
(
X
) is the set of equivalence classes
{
f
1
(
x
1
,...,x
n
)
/f
2
(
x
1
,...,x
n
):
f
1
,f
2
∈ k
[
x
1
,...,x
n
]
,f
2
(0
,...,
0)
=
0
}
.
In other words, the ratios of polynomials such that the denominators always have non-
zero constant coefficient. Similarly, m
P
is the
O
P
-ideal generated by
x
1
,...,x
n
. Since
f
1
(
x
1
,...,x
n
) can be written in the form
f
1
=
c
+
h
(
x
1
,...,x
n
) where
c
∈ k
is the constant
O
P
/
(
x
1
,...,x
n
)
= k
coefficient and
h
(
x
1
,...,x
n
)
∈
m
P
,itfollowsthat
. Hence, m
P
is a
maximal ideal.
n
be a variety over
Exercise 7.1.5
Let
X
⊆ A
k
and let
P
=
(
P
1
,...,P
n
)
∈
X
(
k
). Consider
n
given by
φ
(
x
1
,...,x
n
)
the
translation
morphism
φ
:
X
→ A
=
(
x
1
−
P
1
,...,x
n
−
P
n
).
Show that
φ
(
P
)
=
(0
,...,
0) and that
φ
maps
X
to a variety
Y
that is isomorphic to
X
.
Show further that
O
φ
(
P
)
,
k
(
φ
(
X
)) is isomorphic to
O
P,
k
(
X
)asa
k
-algebra.
We now introduce the notion of singular points and non-singular varieties. These con-
cepts are crucial in our discussion of curves: on a non-singular curve, one can define the
order of a pole or zero of a function in a well-behaved way. Since singularity is a local
property of a point (i.e., it can be defined in terms of
O
P
) it is sufficient to restrict attention
to affine varieties. Before stating the definition we need a lemma.
