Graphics Reference
In-Depth Information
First of all,
V
2
Ã
V
implies that I(
V
) Ã I(
V
2
). On the other hand, since
V
1
is con-
tained in but not equal to
V
there must exist a polynomial f
1
ΠI(
V
1
) - I(
V
). Let f
2
be
any polynomial in I(
V
2
). It follows that f
1
f
2
vanishes on
V
, that is, f
1
f
2
ΠI(
V
). Since
I(
V
) is prime, either f
1
or f
2
belongs to I(
V
). It must be f
2
because f
1
does not by hypoth-
esis. Since f
2
was an arbitrary element of I(
V
2
), we have shown the opposite inclusion
I(
V
2
) Ã I(
V
) and the Claim is proved.
The claim and Theorem 10.8.3(3) imply that
V
=
V
2
and the theorem is proved.
Theorem 10.8.12 and Corollary 10.8.6 lead to
10.8.13. Corollary.
If k is algebraically closed, then the maps I and V define
correspondences
I
¨æ
æ
n
[
]
irreducible
var
ieties of k
prime ideals in k X
,
X
,...,
X
.
æ
12
n
V
that are one-to-one and onto.
While we are on the subject of irreducibility, Theorem 10.8.14 may be an important
theoretical result but not as useful as one would like, because checking the primality of
an ideal is not that easy. Therefore, it is nice to have a theorem like the one below
because it provides a practical means to establishing the irreducibility of many varieties.
Let k be an infinite field. If
V
is a variety in k
n
10.8.14. Theorem.
that admits a
parameterization of the form
(
)
=
(
(
)
(
)
(
)
)
p t
,
t
,...,
t
f t
,
t
,...,
t
,
f
t
,
t
,...,
t
,...,
f
t
,
t
,...,
t
,
12
m
112
m
212
m
n
12
m
where f
i
Πk[t
1
,t
2
,...,t
m
], then
V
is irreducible.
Proof.
Let
()
=Œ
[
{
]
(
(
)
)
I
h
k XXXh p t
,
,...,
,
t
,...,
t
is
the zero polynomial
12
n
12
m
[
]
}
in k t
1
,
t
,...,
t
.
2
m
By Theorem 10.8.12, to prove the theorem in this case, it suffices to show that I(
V
) is
a prime ideal. Let r, s Πk[X
1
,X
2
,...,X
n
] and assume that rs ΠI(
V
). Then (rs)
°
p =
(r
°
p)(s
°
p) is the zero polynomial in k[t
1
,t
2
,...,t
m
] and so either r
°
p or s
°
p is the zero
polynomial. In other words, either r or s belongs to I(
V
) and we are done.
As one example of an application of Theorem 10.8.14, we see that all planes in
R
n
are irreducible because they can be parameterized by linear polynomials. Another
example is a parabola in
R
2
. Later we shall see (Theorem 10.14.6) that Theorem
10.8.14 generalizes to rational functions.
The next step in our program is to generalize Theorem 10.5.13(2) from hyper-
surfaces to arbitrary varieties.
10.8.15. Theorem.
Every variety
V
can be expressed as a finite union of irreducible
varieties.
