Graphics Reference
In-Depth Information
Figure 10.19.
A curve homeomorphic but not iso-
morphic to
R
.
y
V(y
5
- x
2
)
x
It is easy to check that their pullback maps
[]
Æ
[]
[]
Æ
[]
p
*:
k
WV
k
and
s
* :
k
VW
k
are inverses of each other. The map s* basically sends every polynomial F(X,Y,Z) on
V
into the polynomial where Z has been replaced by f(X,Y).
10.13.7. Example.
The variety V(Y
5
- X
2
) »
R
2
is homeomorphic but not isomor-
phic to
R
. See Figure 10.19. The projection to the x-axis clearly defines a homeo-
morphism between it and
R
. The proof that it is not isomorphic to
R
is somewhat
involved. See [CoLO97].
Example 10.13.7 shows that the algebraic concept of isomorphism is stronger than
the topological concept of homeomorphism.
After this overview of polynomial functions we are ready to tackle rational func-
tions. Intuitively, the reader should think of a rational function on a variety as a func-
tion which can be expressed as a quotient p/q of polynomials p and q. Unfortunately,
this intuitive definition runs into lots of technical problems which would require a
lengthy discussion to overcome. For example, such a “function” is not defined at points
where the denominator q vanishes. This in turn would make it tricky to define the com-
position of such functions. There are alternate, more abstract ways to define rational
functions that avoid these difficulties and lead to the basic theorems more quickly. In
the interest of saving time, we shall take one of these approaches so that we can give rig-
orous statements of theorems, even though the author normally prefers to go from the
concrete to the abstract rather than vice versa. We shall sketch how the abstract con-
cepts relate to the intuitive ones as we go along. Our approach follows that of [Shaf94].
Definition.
Let
V
be an irreducible affine variety. The
function field
or
field of rational
functions
of
V
, denoted by k(
V
) is defined to be the quotient field of the coordinate
ring k[
V
] and its elements are called
rational functions
on
V
.
The next proposition shows among other things that our abstract definition of
rational function really is simply another way of expressing the intuitive idea of a quo-
tient of polynomials.
10.13.8. Proposition.
If
V
is an irreducible affine variety, then k(
V
) is a well-defined
field. Furthermore,
