Cryptography Reference
In-Depth Information
of classes under the equivalence relation
f
1
/f
2
≡
f
3
/f
4
if and only if
f
1
f
4
−
f
2
f
3
∈
I
k
(
X
).
In other words,
k
(
X
) is the field of fractions of the affine coordinate ring
k
[
X
] over
k
.
Let
X
be a projective variety. The
function field
is
k
(
X
)
={
f
1
/f
2
:
f
1
,f
2
∈ k
[
X
] homogeneous of the same degree
,f
2
∈
I
k
(
X
)
}
with the equivalence relation
f
1
/f
2
≡
f
3
/f
4
if and only if
f
1
f
4
−
f
2
f
3
∈
I
k
(
X
).
Elements of
k
(
X
) are called
rational functions
.For
a
∈ k
the rational function
f
:
X
→ k
given by
f
(
P
)
=
a
is called a
constant function
.
Exercise 5.4.2
Prove that the field of fractions of an integral domain is a field. Hence,
deduce that if
X
is an affine variety then
k
(
X
) is a field. Prove also that if
X
is a projective
variety then
k
(
X
) is a field.
We stress that, when
X
is projective,
k
(
X
) is not the field of fractions of
k
[
X
] and that
k
[
X
]
⊆ k
(
X
). Also note that elements of the function field are not functions
X
→ k
but
maps
X
→ k
(i.e., they are not necessarily defined everywhere).
2
)
= k
2
)
= k
Example 5.4.3
One has
k
(
A
(
x,y
) and
k
(
P
(
x,y
).
Definition 5.4.4
Let
X
be a variety and let
f
1
,f
2
∈ k
[
X
]. Then
f
1
/f
2
is
defined
or
regular
at
P
if
f
2
(
P
)
=
0. An equivalence class
f
∈ k
(
X
)is
regular
at
P
if it contains some
f
1
/f
2
with
f
1
,f
2
∈ k
[
X
](if
X
is projective then necessarily deg(
f
1
)
=
deg(
f
2
)) such that
f
1
/f
2
is regular at
P
.
Note that there may be many choices of representative for the equivalence class of
f
,
and only some of them may be defined at
P
.
Example 5.4.5
Let
k
be a field of characteristic not equal to 2. Let
X
be the algebraic set
V
(
y
2
2
(
−
x
(
x
−
1)(
x
+
1))
⊂ A
k
). Consider the functions
x
(
x
−
1)
y
1
.
One can check that
f
1
is equivalent to
f
2
. Note that
f
1
is not defined at (0
,
0)
,
(1
,
0) or
(
f
1
=
and
f
2
=
y
x
+
1
,
0). The equivalence class of
f
1
is therefore regular at (0
,
0) and (1
,
0). Section
7.3
gives techniques to deal with these
issues for curves, from which one can deduce that no function in the equivalence class of
f
1
is defined at (
−
1
,
0), while
f
2
is defined at (0
,
0) and (1
,
0) but not at (
−
−
1
,
0).
Exercise 5.4.6
Let
X
be a variety over
k
. Suppose
f
1
/f
2
and
f
3
/f
4
are equivalent functions
on
X
that are both defined at
P
∈
X
(
k
). Show that (
f
1
/f
2
)(
P
)
=
(
f
3
/f
4
)(
P
).
Hence, if
f
is a function that is defined at a point
P
then it makes sense to speak of the
value
of the function at
P
.Ifthevalueof
f
at
P
is zero then
P
is called a
zero
of
f
.
5
5
For curves we will later define the notion of a function
f
having a pole at a point
P
. This notion does not make sense for general
varieties, as shown by the function
x/y
on
A
2
at (0
,
0) for example.
