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(1) V (I + J) = V(I) « V(J).
(2) V (I • J) = V(I) » V(J).
(3) V (I « J) = V(I) » V(J).
(4) We always have V(I : J) … V(I) - V(J), but if k is algebraically closed and I is a
radical ideal, then V(I : J) = V(I) - V(J). (
X
is the closure of the space X.)
Proof. Parts (1)-(3) are straightforward. Part (4) needs the Hilbert Nullstellensatz.
See, for example, [CoLO97].
Up to here we have dealt with affine varieties in this section. We now turn our atten-
tion to projective varieties and show how one can associate ideals to them also. Recall
that to define projective varieties we had to use homogeneous polynomials. Therefore,
it might seem as if the natural ideals to associate to projective varieties are those that
consist of homogeneous polynomials. The problem with that is that the sum of two
homogeneous polynomials may not be homogeneous. The right definition is the fol-
lowing:
Definition. An ideal in k[X 1 ,X 2 ,...,X n ] is said to be homogeneous if it is generated
by a set of homogeneous polynomials.
If V is a projective variety in P n (k), define
Definition.
() =< Π[
]
I
V
f
k X
,
X
,...,
X
f is a homogeneous polynomial t
hat is zero on
V
>
.
12
n
If I is a homogeneous ideal of k[X 1 ,X 2 ,...,X n ], define
{
}
() () () =
n
VI
pP
k f
p 0 for all homogeneous polynomials f
Œ
I .
10.8.20. Proposition.
(1) An ideal I of k[X 1 ,X 2 ,...,X n ] is a homogeneous ideal if and only if I =<f 1 ,f 2 ,
...,f s >, where the f i are homogeneous polynomials.
(2) If I is a homogeneous ideal in k[X 1 ,X 2 ,...,X n ], then ÷ - is a homogeneous ideal.
(3) If V is a projective variety in P n (k), then V(I( V )) = V .
Proof.
The proofs are not hard. Part (1) needs the Hilbert Basis Theorem.
It follows from Proposition 10.8.20(1) that V(I) is a projective variety for all homo-
geneous ideals I. With the definitions above and Proposition 10.8.20 one can estab-
lish a correspondence between projective varieties and radical homogenous ideals
over algebraically closed fields similar to what we had in the affine case. For the details
and some minor required modifications see [CoLO97]. See Exercise 10.8.5 for another
approach to projective varieties and their algebraic counterparts.
This seems like a good time to introduce another concept.
Definition. The Zariski topology of a variety V in k n or P n (k) is the topology defined
by saying that its closed sets are the subvarieties of V . Alternatively, the sets
 
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