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(2)
V
is irreducible if and only if k[
V
] is an integral domain.
(3) Assume that k is algebraically closed. If
V
is a hypersurface V(f), then the ring
[
]
kX X
,
,...,
X
12
n
k[
V
] is isomorphic to the ring
.
If f is irreducible, then k[
V
]
<>
f
is an integral domain.
Proof.
Consider the map
[
]
Æ
[]
j :
kX X
,
,...,
X
k
V
,
12
n
which sends a polynomial to the function it induces. It is an easy exercise to show
that j is a ring homomorphism with kernel I(
V
), which proves (1). The proof of (2)
is Exercise 10.13.2 (see [Shaf94]). The first part of (3) follows from the fact that
and the second part follows from (2).
()
=<>
I
f
Definition.
If
V
is an affine variety, then k[
V
] is called the
coordinate ring
or
ring of
polynomial functions
of
V
.
Because of Theorem 10.13.2(1) we shall feel free to identify k[
V
] with
[
]
kX X
,
,...,
X
12
n
.
()
I
V
In fact, we will think of it as a vector space over k by identifying the constant func-
tions with k.
To every affine variety
V
we have now associated a ring k[
V
], the coordinate ring.
How does this ring behave with respect to maps?
Definition.
Let u :
V
Æ
W
be a polynomial function between affine varieties
V
and
W
. Define
[]
Æ
[]
uk
*:
WV
k
by
()
=
uf fu
*
o
.
The map u* is sometimes called the
pullback map
defined by u.
10.13.3. Theorem.
u* is a well-defined ring homomorphism that is the identity on
the constant functions. Conversely, if h : k[
W
] Æ k[
V
] is any ring homomorphism that
is the identity on the constant functions, then there exists a unique polynomial map
u:
V
Æ
W
, such that h = u*.
Proof.
The first part of the theorem is easy. It is the proof of the converse part that
is interesting. Assume that
V
Õ k
n
and
W
Õ k
m
. The coordinates of k
m
induce m coor-
dinate functions