Cryptography Reference
In-Depth Information
n
be an affine variety over
Lemma 7.1.6
Let X
⊆ A
k
and let P
∈
X
(
k
)
. Then the quo-
tient ring
O
P,
k
(
X
)
/
m
P,
k
(
X
)
is isomorphic to
k
as a
k
-algebra. Furthermore, the quotient
m
P,
k
(
X
)
/
m
P,
k
(
X
)
2
of
O
P,
k
(
X
)
-ideals is a
k
-vector space of dimension at most n.
Exercise 7.1.7
Prove Lemma
7.1.6
.
As the following example shows, the dimension of the vector space m
P,
k
(
X
)
/
m
P,
k
(
X
)
2
carries information about the local geometry of
X
at the point
P
.
2
(
x,y
), m
P
=
Example
7.1.8
Let
X
= A
and
P
=
(0
,
0)
∈
X
(
k
).
We
have m
P
=
(
x
2
,xy,y
2
) and so the
-vector space m
P
/
m
P
has dimension 2. Note that
X
has dimension
k
2.
V
(
y
2
2
, which has dimension 1. Let
P
Let
X
=
−
x
)
⊆ A
=
(0
,
0)
∈
X
(
k
). Then m
P
=
-vector space m
P
/
m
P
. Since
x
y
2
(
x,y
) and
{
x,y
}
span the
k
=
in
k
(
X
) it follows that
∈
m
P
and so
x
0inm
P
/
m
P
. Hence, m
P
/
m
P
is a one-dimensional vector space over
x
=
k
with basis vector
y
.
Consider now
X
V
(
y
2
x
3
)
2
, which has dimension 1. Let
P
=
−
⊆ A
=
(0
,
0). Again,
spans m
P
/
m
P
over
{
x,y
}
k
. Unlike the previous example, there is no linear dependence
among the elements
(as there is no polynomial relation between
x
and
y
having a
non-zero linear component). Hence, m
P
/
m
P
has basis
{
x,y
}
{
x,y
}
and has dimension 2.
V
(
x
4
y
2
)
2
over
Exercise 7.1.9
Let
X
=
+
x
+
yx
−
⊆ A
k
and let
P
=
(0
,
0). Find a basis
-vector space m
P,
k
(
X
)
/
m
P,
k
(
X
)
2
. Repeat the exercise for
X
V
(
x
4
x
3
for the
k
=
+
+
yx
−
y
2
).
Example
7.1.8
motivates the following definition. One important feature of this definition
is that it is in terms of the local ring at a point
P
and so applies equally to affine and projective
varieties.
Definition 7.1.10
Let
X
be a variety (affine or projective) over
k
and let
P
∈
X
(
k
) be point.
dim(
X
) a
n
d is
singular
otherwise.
1
The variety
X
is
non-singular
or
smooth
if every point
P
(
X
)
2
Then
P
is
non-singular
if dim
m
P,
k
(
X
)
/
m
P,
k
=
k
∈
X
(
k
) is non-singular.
Indeed, it follows from the arguments in this section that if
P
∈
X
(
k
) then
P
is
non-singular if and only if dim
k
m
P,
k
(
X
)
/
m
P,
k
(
X
)
2
dim(
X
). The condition of Defi-
nition
7.1.10
is inconvenient for practical computation. Hence, we now give an equivalent
condition (Corollary
7.1.12
) for a point to be singular.
=
n
be a variety defined over
Theorem 7.1.11
Let X
=
V
(
f
1
,...,f
m
)
⊆ A
k
and let P
∈
-vector space
m
P,
k
/
m
2
P,
k
X
(
k
)
. Let d
1
be the dimension of the
k
. Let d
2
be the rank of the
Jacobian matrix
∂f
i
∂x
j
(
P
)
J
X,P
=
.
1
≤
i
≤
m
1
≤
j
≤
n
Then d
1
+
d
2
=
n.
1
The dimension of the vector space m
P,
k
(
X
)
/
m
P,
k
(
X
)
2
is always greater than or equal to dim(
X
), but we do not need this.
