Download A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves

A BRIEF INTRODUCTION TO SHEAVES
ANDREW BAKER
References
[1]
[2]
[3]
[4]
[5]
R. Godement, Topologie algébrique et théorie des faisceaux, 3rd edition, Hermann (1973).
A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 90 (1957), 119–221.
R. Hartshorne, Algebraic geometry, Springer-Verlag (1977).
B. Iversen, Cohomology of sheaves, Springer-Verlag (1986).
D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358 (1988).
1. Presheaves and sheaves
Let C be a (small) category with a terminal object ∗. For example, in the category of sets
Sets we can take any one element set, while in the category of groups we can take Gps ‘the’
trivial group. In the category of rings Rings we could take the trivial ring, while in the category
of augmented k-algebras AugAlgk we could take k.
Let X be a topological space and view its open sets as the objects of a category TX with
morphisms the inclusion maps U ⊆ V .
A C-valued presheaf on X is a contravariant functor F : TX Ã C. This means that for open
sets U, V, W ⊆ X, F(U ), F(V ) ∈ C and if U ⊆ V there is a restriction morphism ρVU : F(V ) −→
W
V W
F(U ) ∈ C(F(V ), F(U )). Moreover, ρU
U = IdF(U ) , and if U ⊆ V ⊆ W then ρU = ρU ρV .
An element s ∈ F(U ) is called a section of F on/over U . Elements of F(X) are called global
sections.
A C-valued presheaf on X is a sheaf if it satisfies the following condition.
S
• For any collection of open sets {Uλ ⊆ X}λ∈Λ with λ∈Λ Uλ = U and sections sλ ∈ F(Uλ )
satisfying
Uµ
λ
ρU
Uλ ∩Uµ sλ = ρUλ ∩Uµ sµ (λ, µ ∈ Λ),
there is a unique s ∈ F(U ) for which
ρU
Uλ s = s λ
(λ ∈ Λ).
C
We denote the categories of C-valued presheaves and sheaves over X by preShC
X and ShX ,
where the morphisms are natural transformations of functors. It is sometimes more convenient
C
to denote these by preCX and CX . Notice that ShC
X is a (full) subcategory of preShX . This
means there is a forgetful functor
C
( )[ : ShC
X Ã preShX ,
F 7−→ F[
which just views a sheaf as a presheaf. In particular, every morphism of sheaves ϕ : F −→ G
becomes a morphism of presheaves ϕ[ = ϕ : F[ −→ G[ by forgetting the sheaf structure.
Theorem 1.1. There is a sheafification functor
C
( )] : preShC
X Ã ShX ,
which is adjoint to ( )[ .
Date: 07/03/2004.
1
F 7−→ F ]
2
ANDREW BAKER
This means that given a morphism of presheaves ϕ : F −→ G[ where G is a sheaf, there is a
unique sheaf morphism ϕ̃ : F ] −→ G with a factorisation
ϕ̃
[
ϕ : F −→ (F ] )[ −→
G.
Equivalently, there is a natural equivalence of sets
C
]
∼
preShC
X (F, G[ ) = ShX (F , G)
for each sheaf G.
Let F be a (pre)sheaf over X. Then for x ∈ X, we define the stalk of F at x to be
Fx = lim F(U ),
−→
x∈U
where the limit is defined using the maps ρVU : F(V ) −→ F(U ) whenever x ∈ U ⊆ V . Of course
we are assuming that such limits exist in the category C. Notice that for each open set V ⊆ X
with x ∈ V there is a natural map ρVx : F(V ) −→ Fx and we set sx = ρVx s for any section
s ∈ F(V ). Also, a morphism of (pre)sheaves ϕ : F −→ G induces a morphism ϕ : Fx −→ Gx in
C for each x ∈ X.
2. The étale space associated to a presheaf
Let X, X̃ be topological spaces and suppose that π : X̃ −→ X is a continuous map which is
locally a homeomorphism. The sheaf of local sections is given by
Γ(U, π) = {s : U −→ X̃ : s continuous, π ◦ s = IdU },
with the obvious restriction maps.
Conversely, a presheaf F on X has an associated étale space XF which as a set is
a
XF =
Fx .
x∈X
The projection map π : XF −→ X is given by π(s) = x for s ∈ Fx . Notice that each section
s ∈ F(U ) gives rise to a map s̃ : U −→ XF for which
s̃(x) = sx ∈ Fx ⊆ XF .
The topology on XF is defined to have as a basis the sets
s̃(U ) (s ∈ F(U ), U ⊆ X open).
Then Γ( , π) is a sheaf associated to π and there is a morphism of presheaves F −→ Γ( , π)[
which this is an isomorphism if and only if F is a sheaf. In fact, F ] ∼
= Γ( , π).
3. Naturality of sheaves
Given a continuous map f : X −→ Y there are two functors f∗ : preShX −→ preShY and
f −1 : preShY −→ preShX defined as follows.
For an open set W ⊆ Y ,
f∗ F(W ) = F(f −1 W ),
with the obvious restriction maps. If F is a sheaf, so is f∗ F. Hence, f∗ : ShX −→ ShY .
Similarly, if U ⊆ X is open, then
f −1 G(U ) =
lim
−→
G(W )
f (U )⊆W
In general, if G is a sheaf, f −1 G need not be a sheaf. Instead we define f −1 : ShY −→ ShX by
setting
f −1 G = (f −1 G[ )] .
A BRIEF INTRODUCTION TO SHEAVES
3
Theorem 3.1. For a continuous map f : X −→ Y , f∗ : ShX −→ ShY and f −1 : ShY −→ ShX
are adjoint functors, i.e., there is a natural equivalence
C −1
∼
ShC
F, G).
Y (F, f∗ G) = ShX (f
4. Sheaves with algebraic structure
Sheaves are often equipped with algebraic structure. For example, the category C might be
one of the following:
•
•
•
•
•
•
Sets, the category of sets;
Gps, the category of groups;
AbGps, the category of abelian groups;
ModR , the category of modules over a commutative ring R;
Rings, the category of rings or CommRings, the category of commutative rings;
Algk , the category of k-algebras over a unital ring k or CommAlgk , the category of
commutative k-algebras.
If suitable colimits exist in C then each stalk Fx (x ∈ X) is in C. This is true for all of the
above examples.
Important examples of sheaves of algebras are provided by the following.
The sheaf OX,k of continuous k-valued functions X −→ k where k is a topological ring. For
example, we might take k = R, C, Zp , Qp or a ring with discrete topology.
sm is the sheaf of smooth functions on M , where k = R, C.
If M is a smooth manifold then OM,k
hol = O hol is the sheaf of holomorphic functions
If M is a complex (analytic) manifold then OM
M,C
on M .
alg
is the sheaf of regular functions on X.
If X is an affine variety over a field k, then OX,k
5. Ringed spaces
A topological space X together with a sheaf of rings O forms a ringed space and algebraic
geometry is in many respects the study of such objects (X, O). Indeed they form a category
where a morphism (X, OX ) −→ (Y, OY ) is a continuous map f : X −→ Y together with a
morphism of ring sheaves OY −→ f∗ OX over Y , or equivalently a morphism f −1 OY −→ OX
(see Theorem 3.1).
A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with
maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on
each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism
h : R −→ S for which h−1 mS = mR ).
Given a ringed space (X, OX ) and a continuous map g : Z −→ X, g −1 OX is a ring sheaf over
Z. If (Z, OZ ) is a ringed space then OZ is an algebra sheaf over g −1 OX in the sense that there
are compatible ring homomorphisms g −1 OX (W ) −→ OZ (W ) for open sets W ⊆ Z.
If (X, OX ) is a ringed space, a module sheaf over (X, OX ) is a sheaf M which has compatible
module structures OX (U ) × M(U ) −→ M(U ) for each open set U ⊆ X.
Such a module sheaf M is called locally free if for each x ∈ X there is an open neighbourhood
U 3 x with M(U ) a free module over OX (U ). Such a locally free module sheaf is also called a
vector bundle.
Given a morphism of ringed spaces g : (Z, OZ ) −→ (X, OX ) and a module sheaf M over
(X, OX ), we can form g ∗ M = OZ ⊗g−1 OX M to be the sheaf associated to the presheaf
W Ã OZ (W )
If M is locally free then so is g ∗ M.
⊗
g −1 OX (W )
M(W ).
4
ANDREW BAKER
We denote the category of module sheaves over (X, OX ) by ModOX . A morphism of
ringed spaces f : (X, OX ) −→ (Y, OY ) induces two functors f∗ : ModOX −→ ModOY and
f ∗ : ModOY −→ ModOX .
Theorem 5.1. For a morphism of ringed spaces f : (X, OX ) −→ (Y, OY ), f∗ : ModOX −→
ModOY and f ∗ : ModOY −→ ModOX are adjoint functors, i.e., there is a natural equivalence
ModO (f ∗ M, N ) ∼
= ModO (M, f∗ N ).
X
Y
6. Homological algebra in categories of sheaves
Now take C = ModR for some unital ring R. Given a morphism of presheaves ϕ : F −→
G over X taking values in ModR , we can consider the presheaf ker ϕ given by ker ϕ(U ) =
ker ϕ : F(U ) −→ G(U ). If ϕ is a morphism of sheaves then ker ϕ is a sheaf, which we denote
Ker ϕ.
Similarly, we can define Im ϕ to be the presheaf for which Im ϕ(U ) = Im ϕ : F(U ) −→ G(U ).
If ϕ is a morphism of sheaves then we have the associated sheaf Im ϕ = Im ϕ] .
We can also define cokernels. For a morphism of presheaves we set
coker ϕ(U ) = G(U )/ Im ϕ : F(U ) −→ G(U ),
while for a morphism of sheaves we set Coker = coker ϕ] .
There is a constant sheaf 0 given by 0(U ) = 0 (the trivial R-module). It now makes sense
ϕ
θ
to consider a sequence of morphisms of sheaves of R-modules F −
→G−
→ H and define it to be
exact if Ker θ = Im ϕ.
Proposition 6.1. The sequence F −→ G −→ H of morphisms of sheaves of R-modules over
X is exact if and only if the associated sequence of R-modules Fx −→ Gx −→ Hx is exact for
every x ∈ X.
We can extend the above to a sequence of sheaves and morphisms
ϕs−1
ϕs+1
δ
s
· · · −−−→ Fs−1 −→
Fs −→ Fs+1 −−−→ · · ·
and we say that this is exact if for each s, Im ϕs−1 = Ker ϕs .
7. Injective resolutions of sheaves
Let F be a sheaf of R-modules on X. For each x ∈ X, the stalk Fx can be embedded in
an injective R-module using a monomorphism of R-modules j : Fx −→ Jx . For each open set
Q
U ⊆ X, the product x∈U Jx is an injective R-module. Now the presheaf J given by
Y
J (U ) =
Jx
x∈U
with the obvious restriction maps is in fact a sheaf. Moreover, for each open set U , we have an
R-homomorphism
jU : F(U ) −→ J (U ); s 7−→ (sx )x∈U .
This gives rise to a morphism of sheaves j : F −→ J . Then by Proposition 6.1 and the fact
that Jx = Jx for each x ∈ X, we see that j : F −→ J has Ker j = 0, hence is a monomorphism.
Proposition 7.1. The sheaf J is an injective object in the category of R-modules over X.
Proof. Injectivity is expressed by the diagram
0
/M
θ
² ~|
J
ϕ
|
|
/N
|
∃ θ̃
A BRIEF INTRODUCTION TO SHEAVES
5
in which all the arrows are morphisms of sheaves of R-modules on X.
The proof uses by the fact that each J (U ) is an injective R-module.
¤
Proposition 7.2. Every sheaf of R-modules F on X admits a monomorphism F −→ J where
J is injective in the category of sheaves of R-modules over X.
More generally, F admits a resolution
0 −→ F −→ J 0 −→ J 1 −→ · · · −→ J s −→ · · ·
where each J s is injective in the category of sheaves of R-modules over X.
As usual, injective resolutions can be used to calculate cohomology. Namely, give a resolution
0 −→ F −→ J 0 −→ J 1 −→ · · · −→ J s −→ · · ·
of F by injective sheaves of R-modules, we can form the complex
δ
δ
δs−1
δ
0
1
s
0 −→ J 0 (X) −→
J 1 (X) −→
· · · −−−→ J s −→
···
and then define the cohomology groups of the sheaf F on X to be
H s (X, F) = ker δs / Im δs−1
(s > 0).
As usual, this is independent of the actual injective resolution used.
Theorem 7.3. For each sheaf F of R-modules on X, H 0 (X, F) = F(X).
Given a morphism ϕ : F −→ G of sheaves of R-modules on X, there are induced homomorphisms ϕ∗ : H s (X, F) −→ H s (X, F) (s > 0), and when s = 0, ϕ∗ is the evident map on global
sections.
Given an exact sequence of sheaves of R-modules
(7.1)
0 −→ F −→ G −→ H −→ 0,
there is a long exact sequence
0 −→ H 0 (X, F) −→ H 0 (X, G) −→ H 0 (X, H) −→ H 1 (X, F) −→ · · ·
−→ H s (X, F) −→ H s (X, G) −→ H s (X, H) −→ H s+1 (X, F) −→ · · ·
which is natural in the exact sequence (7.1).
There is a weaker notion of ‘relatively injective’ sheaves. A sheaf on X is flasque or flabby if
for every open U ⊆ X, the restriction map ρX
U is surjective. Every injective sheaf of R-modules
s
is flabby. Moreover, for a flabby sheaf F, H (X, F) = 0 when s > 0.
Proposition 7.4. The cohomology H ∗ (X, ) of a sheaf of R-modules F on X can be calculated
by taking a resolution
0 −→ F −→ J 0 −→ J 1 −→ · · · −→ J s −→ · · ·
where each J s is flabby, then taking the cohomology of the complex
0 −→ J 0 (X) −→ J 1 (X) −→ · · · −→ J s (X) −→ · · · ,
i.e., H s (X, F) = ker J s (X) −→ J s+1 (X)/ Im J s−1 (X) −→ J s (X).
There is a canonical flabby resolution that is often encountered. For this we take
Y
F 0 (U ) =
Fx
x∈U
F 0 (U ).
with the obvious embedding F(U ) −→
Then F 0 is a flabby sheaf, moreover there is an
exact sequence
0 −→ F −→ F 0 −→ F 0 −→ 0
6
ANDREW BAKER
where F 0 is the cokernel sheaf. We can now repeat this construction to form the flabby sheaf
Y
F 0x
F 1 (U ) =
x∈U
and an exact sequence
0 −→ F 0 −→ F 1 −→ F 1 −→ 0.
This construction can now be continued to form a resolution
0 −→ F −→ F 0 −→ F 1 −→ · · · −→ F s −→ · · ·
by flabby sheaves with exact sequences
0 −→ F s−1 −→ F s −→ F s −→ 0.
H ∗ (X, F) can be calculated using this resolution.
Another type of cohomology is associated to a continuous map f : X −→ Y . Given an
injective resolution
0 −→ F −→ J 0 −→ J 1 −→ · · · −→ J s −→ · · ·
of the sheaf F on X, we may form the complex
0 −→ f∗ J 0 −→ f∗ J 1 −→ · · · −→ f∗ J s −→ · · ·
whose cohomology is Rs f∗ (F) (s > 0), the higher direct images of F. In fact, R0 f∗ (F) = f∗ F
and there are properties analogous to those of Theorem 7.3. There is another interpretation of
these functors.
For each s > 0, consider the following presheaf on Y :
V 7−→ H s (f −1 V, F|f −1 V ),
where the inclusion map jV : f −1 V −→ X is used to form F|f −1 V = jV−1 F. The associated sheaf
agrees with Rs f∗ (F). In particular, if f = IdX , then Rs Id∗ (F) = 0 for s > 1 and R0 Id∗ (F) = F.
If F is flabby then Rs f∗ (F) when s > 1. There is also a Leray spectral sequence
p
q
p+q
Ep,q
(X, F).
2 = H (Y, R f∗ F) =⇒ H