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Transcript
Chapter 1
Affine Schemes
1.1
Spectrum of a ring
Thus far, we have described classical (affine) algebraic geometry over an algebraically closed field. For lots of reasons, coming from number theory over even
classical geometry it is useful to work over arbitrary fields or even more general
rings. In the last chapter, we showed that reduced affine algebras correspond to
affine varieties. Here we extend the picture to work with arbitrary commutative
rings. Given such a ring R, we can consider the set Max R of maximal ideals
as before, but we run into a fundamental problem. We want this construction
to be functorial, if f : R ! S is a homomorphism of affine algebras of an algebraically closed field, then by the Nullstellensatz the preimage of a maximal
ideal in S is a maximal ideal of R. This is no longer true for a homomorphism
of general rings. What is true is that the preimage of maximal ideal, or more
generally a prime ideal, is prime. This suggests changing the emphasis to the
set Spec R of prime ideals of R. This is called the (prime) spectrum of R. This
contains Max R and much more. For example, if R = k[x, y], where k is a field,
then Spec R consists of the union of {0}, Max R, and the set of principal ideals
associated to irreducible polynomials.
Given an ideal I ⇢ R, let
V (I) = {p 2 Spec R | p ◆ I}
This is consistent with the earlier notation, in the sense that the intersection of
the above V (I) with Max R agrees with the previous.
Theorem 1.1.1.
1. V (0) = Spec R, V (1) = ;.
2. Given an arbitrary family of ideals Ij , V (
P
3. If I and J are ideals, V (I) \ V (J) = V (IJ)
7
Ij ) =
T
V (Ij ).
Recall that a topology on a set X consists of the collection of subsets call
open sets such that
1. X and ; are open.
2. An arbitrary union of open sets is open.
3. A finite intersection of open sets is open.
A subset is closed if and only if its complement is open.
Corollary 1.1.2. The collection {Spec R V (I)} defines a topology on Spec R,
such that the closed sets are precisely the sets V (I).
This is called the Zariski topology. Unlike the topologies that you encounter
in analysis, this is almost never Hausdor↵.
Exercise 7.
1. Let X = Spec R, where R is an integral domain. Then the zero ideal 0 is
prime, and therefore defines a point 0 2 X. Show that the closure of {0}
is X. Conclude that if X contains at least one other point, then X is not
Hausdor↵ (nor even T1 if you know what that means).
2. Given f 2 R, D(f ) = Spec R V (f ) is an open set, called a basic open
set. Show that every open set is a union of basic open sets.
S
3. Prove that X = Spec R is quasi-compact which means that if X = Ui
for open sets Ui , then X is a union of a finite number of them. (This
condition is just called compactness in American terminology; here we are
using French terminology where compact = quasi-compact + Hausdor↵.)
4. Given a homomorphism f : R ! S, let F : Spec S ! Spec R be defined
by F (p) = f 1 p. Prove that this is continuous with respect to the Zariski
topologies. (Recall that F is continuous , the preimage of open sets are
open , the preimage of closed sets are closed.)
Exercise 4 and a bit more work shows that R 7! Spec R is a contravariant
functor from the category of commutative rings to the category of topological
spaces.
1.2
Localization
Given a set S ⇢ R which is multiplicative in the sense it is closed under multiplication, there exists a universal ring R[S 1 ] = S 1 R and homomorphism
u : R ! S 1 R such that elements u(S) are invertible. The elements can be
represented by fractions r/s, s 2 S [ {1}, subject to the relation that
r/s = r0 /s0 , 9t 2 S, t(rs0
8
r0 s) = 0
There two cases of particular interest, if S = {f n }, then Rf = R[1/f ] is obtained
by formally inverting f . If p 2 Spec R, then S = R p is multiplicative, we set
Rp = S 1 R. This should not be confused with the quotient R/p which has the
e↵ect of turning any ideal contained in p into the zero ideal. The localization
Rp does the opposite. It turns an ideal strictly containing p into the unit ideal.
Proposition 1.2.1. Let pRp denote the ideal generated by p. This is the unique
maximal ideal of Rp ; in particular, Rp is a local ring.
Proof. If 1 2 pRp , then 1 = r/f with r 2 p, f 2
/ p. Therefore 9t 2
/ p, tf = rt,
which is impossible because tf 2
/ p while rt 2 p. Thus pRp is a proper ideal.
Any element of the complement of pRp is a unit, and this implies that it is the
unique maximal ideal.
Given k[x1 , . . . xn ], with k algebraically closed, we can idenfity Max k[x1 , . . . xn ] =
Ank . The elements of k[x1 , . . . xn ] can be viewed as regular functions on Ank . We
want to extend this viewpoint to arbitrary rings. We will view an element of R
as a generalized function on Spec R. To emphasize this, we set O(Spec R) = R,
which is the ring of “regular functions” on Spec R. For regular functions on
a basic open D(f ), we want to allow poles on the complement of D(f ). So
we set O(D(f )) = R[1/f ]. The natural homomorphism R ! R[1/f ] can be
understood as restriction of a regular function on X to a regular function on
D(f ).
In general, determining regular functions O(U ) for a nonbasic open set U is
more subtle. It is determined by the following rules.
1. If V ⇢ U is an open subset, we can restrict an element of O(U ) to O(V ).
This gives a homomorphism. If W ⇢ V , then restriction from U to W is
the same as restricting from U to V and then from V to W .
2. If U = [Ui is an open cover, then f 2 O(U ) is uniquely determined by the
restrictions f |Ui . Given a family fi 2 O(Ui ) such that fi |Ui \Uj = fj |Ui \Uj ,
there exists f 2 O(U ) such that fi = f |Ui .
The first condition is that the collection O(U ) together with restrictions
forms a presheaf of rings. The second condition is that this forms a sheaf. We
will say more about this later. For now, we summarize the state of a↵airs:
Theorem 1.2.2. There exists a sheaf of rings O on Spec R such that O(D(f )) ⇠
=
R[1/f ]
S
Corollary 1.2.3. Suppose that U = i2I D(fi ) ⇢ Spec R is open. Then
⇢✓ ◆
Y
ri
⇠
| 9nij , (fi fj )nij (ri fj rj fi ) = 0 ⇢
R[1/fi ]
O(U ) =
fi i2I
i2I
The corollary implies that two di↵erent
coverings should yield isomorphic
S
rings. In particular, if X = Spec R = D(fi ), then we should have
9
Corollary 1.2.4. The sequence
Y
⇢ Y
0!R!
R[1/fi ] !
R[1/fi fj ]
i
i6=j
is exact, where the ⇢ is the product of natural maps R ! R[1/fi ], and the product of R[1/fi ] ! R[1/fi fj ] (i < j) minus the product of R[1/fi ] ! R[1/fi fj ] (i >
j).
Although we won’t give a proof of the theorem, it is instructive to give a
direct proof of the last corollary, which uses similar ideas.
Proof. Using the quascompactness statement proved in an earlier exercise, we
can assume there are finitely many fi ’s. Suppose that
T ⇢(r) = 0. This
S means that
fini r = 0. Let n = max ni , then fin r = 0. Since V (fin ) = X
D(fi ) = ;,
the ideal I = (f1n , f2n , . . .) must equal 1. Otherwise we would have a maximal
ideal containing
I which would contradict V (I) = ;. Therefore we have gi 2 R
P
such that
gi fin = 1. This implies that
X
r=
gi fin r = 0
Thus ⇢ is injective.
Next suppose that ((ri /fi )) = 0. Therefore
(fi fj )n (ri fj
rj f i ) = 0
for some n (which can be chosen independent of i, j). Setting bi = fin ri and
hi = gin+1 , we can rewrite the above equation as
bi h j = b j h i
S
P
NowP
we have D(hi ) = X, which implies 1 =
ai hi for some ai 2 R. Set
r=
ai bi . Then
X
X
rhj =
a i bi h j =
a i hi b j = b j
or
r = bj /hj = rj /fj
This proves ker
1.3
= im ⇢ as required.
A little sheaf theory
Sheaves were introduced in passing in the last section. Let us take a more careful
look. A presheaf of sets, groups rings... on a topological space X consists of
an assignment of F(U ) of a set, group... to each open U ⇢ X and a map or
homorphism ⇢U V : F(U ) ! F(V ) (called restriction) for each pair V ✓ U , such
that
10
1. ⇢U U = id
2. If W ⇢ V ⇢ U , then ⇢U W = ⇢V W
⇢U V .
In practice, we write fV instead of ⇢U V (f ). There are many examples which
have nothing to do with algebraic geometry.
Example 1.3.1. Let X be a topological space, and fix a set, group... Y . Let
F unY (U ) be the set of all functions from U to Y with restriction having the
usual meaning. This forms a presheaf of sets, groups...
Example 1.3.2. With X and Y as above, let Y (U ) denote the set of constant
functions from U to Y .
Example 1.3.3. With X a topological space let C(U ) denote the set of continuous functions from U to C. This is a presheaf of rings.
The first and third example have an additional desirable property worth
noting. A presheaf
F on X is called a sheaf, if for any open set U and open
S
cover U = Ui . Given a collection fi 2 F(Ui ) such that fi |Ui \Uj = fj |Ui \Uj ,
there exists a unique f 2 F(U ) with f |Ui = fi . The second example is not
sheaf with some trivial exceptions, because if U = U1 [ U2 is a dsjoint union,
then we can assign fi (x) = yi on Ui and unless y1 = y2 , the resulting function
is not constant. In general, a presheaf of functions is a sheaf if the definining
conditions are local. For some examples from calculus:
Exercise 8. Show that the presheaf on R, which assigns to each open set U
the ring of di↵erentiable functions on it, is a sheaf. Is the presheaf of square
integrable or L2 functions a sheaf ?
There is a simple way to turn the previous nonsheaf into a sheaf:
Example 1.3.4. Let Y + (U ) denote the set of locally constant functions from
U to Y . This is a sheaf called the sheafification,
The process of going from Y to Y + works for any sheaf. In other words, there
is a canonical way to turn a presheaf into a sheaf. Given a pair of presheaves (or
in particular sheaves) F, G a morphism is a collection of maps ⌘U : F(U ) ! G(U )
such that ⌘U (f )|V = ⌘V (f |V ). We say that this is an isomorphism if each ⌘U is
a bijection of sets or isomorphism of groups....
Theorem 1.3.5. Given a presheaf F, there exists a sheaf F + with a morphism
F ! F + such any other morphism to a sheaf F ! G factors uniquely through
F +.
As an example where this occurs, let F, G be a pair of sheaves of abelian
groups, with a morphism ⌘ : F(U ) ! G(U ) which is injective for each U . Then
we can view F(U ) ⇢ G(U ), and we refer to F as a subsheaf of G. The quotient
Q(U ) = G(U )/F(U ) is a presheaf which is usually not a sheaf. We define the
quotient sheaf by G/F = Q+ .
11
Before constructing the sheafification, we need one more notion. Given a
presheaf F on X, x 2 X and fi 2 F(Ui ) where x 2 Ui , define f1 ⇠ f2 if their
restrictions coincide in some neighbourhood of x contained in U1 \ U2 . This
is an equivalence relation. The equivalence class of f is called the germ of f
at x. Let us denote this by gx (f ). The stalk Fx is the set of all germs gx (f ).
Alternatively, Fx = lim F(U ), as U varies over open neigbourhoods of x. We
!
define
Y
F + (U ) ⇢
Fx
x2U
+
where (fx ) 2 F (U ) if for each x, there exists a neighbourhood x 2 V ⇢ U , a
section 2 F(V ) such that fy = gy ( ) for all y 2 V .
1.4
Affine Schemes
Given a ring R, we constructed a topological space Spec R. This is however a
fairly crude invariant. There is no hope of recovering R from its spectrum. On
the other hand using the sheaf O constructed earlier we can recover R using the
formula O(Spec R) = R. The pair (Spec R, O) is called the the affine scheme
associated to R. In practice, we usually just abbreviate this as Spec R.
12