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In-Depth Information
Claim 1.
V(I(V(
I
))) Õ V(
I
)
Let
p
ΠV(I(V(
I
))). Then f(
p
) = 0 for all f ΠI(V(
I
)). Since
I
is clearly contained in
I(V(
I
)),
p
must belong to V(
I
) and Claim 1 is proved.
Claim 2.
V(
I
) Õ V(I(V(
I
)))
Let
p
ΠV(
I
). Given an arbitrary f ΠI(V(
I
)), f(
p
) = 0 by definition and so Claim 2
is obvious. The two inclusions in Claim 1 and 2 prove the equality in (3).
Next, we state another of the really important theorems of algebraic geometry, the
Hilbert Nullstellensatz. This theorem gets used in many places and in particular we
shall need it. We state two variants. The proofs are not that difficult but still too long
to present here.
10.8.4. Theorem.
(The Hilbert Nullstellensatz: weak form) Let k be an algebraically
closed field. If I is a proper ideal in k[X
1
,X
2
,...,X
n
], then there is at least one point
on which all polynomials of I vanish.
Proof.
See [CoLO97].
10.8.5. Theorem.
(The Hilbert Nullstellensatz) Let k be an algebraically closed field
and let f, f
1
, f
2
,..., and f
k
be polynomials in k[X
1
,X
2
,...,X
n
]. If f vanishes on all the
common zeros of the f
i
, then some power of f is a linear combination of the f
i
, that
is, for some m > 0,
m
f
=+ ++
af
af
...
af
.
11
2 2
kk
In other words, if I is an ideal in k[X
1
,X
2
,...,X
n
] then I(V(I)) =÷
-
.
Proof.
See [CoLO97]. The proof uses Theorem 10.8.4.
An immediate consequence of the Nullstellensatz is that we can state the corre-
spondence between varieties and ideals more precisely.
10.8.6. Corollary.
If k is algebraically closed, then the maps I and V define corre-
spondences
I
¨æ
æ
n
[
]
var
ieties of k
radical ideals in k X
,
X
,...,
X
.
æ
12
n
V
that are one-to-one and onto.
Proof.
We already know from Theorem 10.8.3(3) that V(I(
V
)) =
V
. To show that
I(V(I)) = I for all radical ideals I we simply observe that I(V(I)) =÷
-
by the Nullstel-
lensatz and ÷
-
= I whenever I is a radical ideal.
Another consequence of the Nullstellensatz is an algebraic characterization of
points.
