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Chapter 8
Introducing Algebraic
Geometry
(Commutative Algebra 2)
We will develop enough geometry to allow an appreciation of the Hilbert Nullstellensatz,
and look at some techniques of commutative algebra that have geometric signiFcance. As
in Chapter 7, unless otherwise speciFed, all rings will be assumed commutative.
8.1 Varieties
8.1.1 Defnitions and Comments
We will be working in
k
[
X
1
,...,X
n
], the ring of polynomials in
n
variables over the Feld
k
. (Any application of the Nullstellensatz requires that
k
be algebraically closed, but we
will not make this assumption until it becomes necessary.) The set
A
n
=
A
n
(
k
) of all
n
tuples with components in
k
is called
aﬃne
n
space
.I
f
S
is a set of polynomials in
k
[
X
1
n
], then the zeroset of
S
, that is, the set
V
=
V
(
S
)o
fa
l
l
x
∈
A
n
such that
f
(
x
) = 0 for every
f
∈
S
, is called a
variety
. (The term “aﬃne variety” is more precise,
but we will use the short form because we will not be discussing projective varieties.)
Thus a variety is the solution set of simultaneous polynomial equations.
If
I
is the ideal generated by
S
, then
I
consists of all Fnite linear combinations
∑
g
i
f
i
with
g
i
∈
k
[
X
1
n
] and
f
i
∈
S
. It follows that
V
(
S
)=
V
(
I
), so every variety is the
variety of some ideal. We now prove that we can make
A
n
into a topological space by
taking varieties as the closed sets.
8.1.2 Proposition
(1) If
V
α
=
V
(
I
α
) for all
α
∈
T
, then
T
V
α
=
V
(
S
I
α
). Thus an arbitrary intersection of
varieties is a variety.
1
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CHAPTER 8. INTRODUCING ALGEBRAIC GEOMETRY
(2) If
V
j
=
V
(
I
j
),
j
=1
,...,r
, then
S
r
j
=1
V
j
=
V
(
{
f
1
···
f
r
:
f
j
∈
I
j
,
1
≤
j
≤
r
}
). Thus a
Fnite union of varieties is a variety.
(3)
A
n
=
V
(0) and
∅
=
V
(1), so the entire space and the empty set are varieties.
Consequently, there is a topology on
A
n
, called the
Zariski topology
, such that the
closed sets and the varieties coincide.
Proof.
(1) If
x
∈
A
n
, then
x
∈
T
V
α
i± every polynomial in every
I
α
vanishes at
x
i±
x
∈
V
(
S
I
α
).
(2)
x
∈
S
r
j
=1
V
j
i± for some
j
, every
f
j
∈
I
j
vanishes at
x
i±
x
∈
V
(
{
f
1
f
r
:
f
j
∈
I
j
for all
j
}
).
(3) The zero polynomial vanishes everywhere and a nonzero constant polynomial van
ishes nowhere.
♣
Note that condition (2) can also be expressed as
∪
r
j
=1
V
j
=
V
r
Y
j
=1
I
j
=
V
(
∩
r
j
=1
I
j
)
.
[See (7.6.1) for the deFnition of a product of ideals.]
We have seen that every subset of
k
[
X
1
,...,X
n
], in particular every ideal, determines
a variety. We can reverse this process as follows.
8.1.3 Defnitions and Comments
If
X
is an arbitrary subset of
A
n
, we deFne the
ideal of
X
as
I
(
X
)=
{
f
∈
k
[
X
1
n
]:
f
vanishes on
X
}
. By deFnition we have:
(4) If
X
⊆
Y
then
I
(
X
)
⊇
I
(
Y
); if
S
⊆
T
then
V
(
S
)
⊇
V
(
T
).
Now if
S
is any set of polynomials, deFne
IV
(
S
)as
I
(
V
(
S
)), the ideal of the zeroset
of
S
; we are simply omitting parentheses for convenience. Similarly, if
X
is any subset
of
A
n
, we can deFne
VI
(
X
),
IVI
(
X
),
VIV
(
S
), and so on. ²rom the deFnitions we
have:
(5)
(
S
)
⊇
S
;
(
X
)
⊇
X
.
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 Spring '11
 sdd
 Algebra, Geometry

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