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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
 
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