Basic Constructions

Section 2: Basic Constructions

2 Basic Constructions

When F is a sheaf, it is legitimate to call elements of F(U) sections of F over an

open set U , since they can be identied with sections of the associated étale space.

Continuous sections of the étale space make sense, on the other hand, over any subspace

of X.

2.1. Proposition. If K is a closed subspace of a paracompact topological space X,

then any section over K of a sheaf F on X is the restriction to K of a section of F over

a neighbourhood of K.

Proof. In fact, any section s over K is dened by open sets U

i

of X that cover K and

elements s

i

∈ F(U

i

). Since K is also paracompact, we may assume (by passing to a

renement, if necessary) that the covering is locally nite. Let U =

∪

U

i

and {V

i

} be a

shrinking of this covering. Choose for every x ∈ U, an open neighbourhood M

x

of x as

in Lemma 1.13. Denote by I

x

the set of all i ∈ I such that x ∈ V

i

. Then our choice of

M

x

is equivalent to saying that i belongs to I

x

if and only if M

x

intersects V

i

. Consider

the subset W = {x ∈ U : (s

i

)

x

is independent of i for all i ∈ I

x

}. For any y ∈ M

x

,

we have I

y

⊂ I

x

. For, if i ∈ I

y

, then y ∈ V

i

and hence M

x

which is a neighbourhood

of y has nonempty intersection with V

i

. This means that i ∈ I

x

. Clearly, the set

{y ∈ M

x

: (s

i

)

y

is independent of i ∈ I

x

} is an open set containing x and contained in

W . Hence W is open. On the other hand we have the inclusion K ⊂ W . It is now clear

that the s

i

actually give a section of F over W .

2.2. Remark. Let us consider the sheaf of continuous functions on R

n

for example. A

section of the corresponding étale space over a closed set

K

is the same as a continuous

function in a neighbourhood of K with the understanding that two such continuous

functions are to be considered equivalent if they coincide in a neighbourhood of K. Any

such ‘germ’ gives, on restriction to K, a continuous function on K. On the other hand

any continuous function on K can be extended to some neighbourhood of K. Thus we

have a surjection of the set of sections over K of the sheaf of continuous functions into

the set of continuous functions on K. But this is not injective, even when K consists of

a single point. For in this case, a section is simply an element of the stalk at the point,

which consists of germs of continuous functions at the point.

2.3. Inverse images. We used the construction of E(F) as a means to pass from a

presheaf to a sheaf. However, even when one starts with a sheaf the construction of the

étale space is useful in some applications. One such is the notion of the inverse image of

a sheaf. Let f : X → Y be a continuous map of topological spaces. If F is a sheaf on Y ,

we seek to dene a sheaf f

−1

F on X. Let E(F) be the étale space of F. Then form the

bre product of the map f : X → Y and the map π : E(F) → Y . (It is the subspace of

17

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

the topological space X ×E(F) consisting of points (x, a) such that f(x) = π(a).) This

space comes with a natural continuous map into X. The sheaf of continuous sections of

this space is called the inverse image of F by the map f.

In the particular case when X ⊂ Y is a subspace, the inverse image is also called the

restriction of F to X. We sometimes use the notation F|X for this restriction.

2.4. Glueing together. Let (U

i

) be an open covering of a space Y and for each i,

let F

i

be sheaves on U

i

. Then we wish to glue all these together and obtain a sheaf

on the whole of Y . For this we need some glueing data. If the F

i

are all restrictions

to U

i

of the same sheaf F on Y , then the restrictions of F

i

and F

j

to U

i

∩ U

j

are

the same as direct restrictions of F to U

i

∩ U

j

. So, we assume as data, isomorphisms

m

ij

: F

j

|U

i

∩ U

j

→ F

i

|U

i

∩ U

j

. Actually we need more, namely compatibility of these

isomorphisms. Consider the set U

i

∩ U

j

∩ U

k

. We have restrictions to it of the sheaves

F

i

, F

j

and F

k

. Besides, the isomorphisms m

ij

, m

jk

and m

ik

restrict to isomorphisms

(two by two) of these three sheaves on U

i

∩ U

j

∩ U

k

as well. We will use the same

notation for these restrictions. Then the compatibility condition that we have in mind

is that they should satisfy

m

ij

◦ m

jk

= m

ik

One can easily verify that given such data as above, one can glue the sheaves F

i

together and obtain a sheaf F on the whole of Y with natural isomorphisms of F|U

i

with F

i

.

Actually, in a certain sense, the construction involved in glueing is the converse of

the construction of the inverse image. If X is the topological union of the spaces U

i

,

then it is clear that the F

i

build a sheaf

˜

F on it, and what is needed is a sheaf on Y

whose inverse image under the natural map X → Y is

˜

F.

2.5. Remark. Regarding our glueing construction above, we wish to make the fol-

lowing remark. Let F be a sheaf on X. Let (U

i

) be an open covering and let G

i

be

subsheaves of

F|

U

i

, for all

i

. Then, in order to glue the

G

i

together, we only need to

check that G

i

|U

i

∩ U

j

is the same subsheaf of F|U

i

∩ U

j

as G

j

|U

i

∩ U

j

. Then we can,

not only glue them together, but also get a homomorphism of G into F which makes it

a subsheaf.

2.6. Denition. Let F be a sheaf on a space X. Let f : X → Y be a continuous

map. Then one can dene a sheaf on Y called the direct image of F by f as follows. To

any open set U in Y , associate F(f

−1

(U)). It is easy to check that this denes a sheaf.

We will denote this by f

∗

(F).

This is related to the inverse image very closely. In fact, let F be a sheaf on X in the

above situation, and G a sheaf on Y . We may consider on the one hand, the direct image

f

∗

(F) and on the other, the inverse image f

−1

(G). Then one can see easily that there is a

natural bijection between homomorphisms G → f

∗

(F) and f

−1

(G) → F. Indeed, let us

18

Section 2: Basic Constructions

start with a homomorphism T : G → f

∗

(F). This gives, for every open subset U ⊂ Y a

homomorphism T (U ) : G(U) → (f

∗

F)(U ) = F(f

−1

(U)). If x ∈ f

−1

(U), then we have a

natural map F(f

−1

(U)) → F

x

. Composing with T (U ), we get a homomorphism G(U) →

F

x

. It is obvious that as U varies over neighbourhoods of f (x), this is compatible with

restrictions, and so induces a map G

f(x)

= f

−1

(G)

x

→ F

x

. This is easily checked to be

continuous on the étale space and hence gives a homomorphism of f

−1

(G) into F. It is

this association that gives the bijection, as claimed.

2.7. Modules over a sheaf of algebras. Before we leave this preliminary account

of sheaves and move on to dierential manifolds, we would like to introduce one more

notion which is very useful in our context. As we have observed, the sheaf of continuous

functions on a topological space, that of dierentiable functions in R

n

, that of holomor-

phic functions in a domain in C

n

, etc. are all sheaves of algebras. Let then A be a sheaf

of algebras over a topological space X. On the other hand, let M be a sheaf of abelian

groups. Then we say that M is a sheaf of modules over A, or simply an A-module

if for every U, open in X, the abelian group M(U) comes provided with a structure

of an A(U)-module in such a way that the restriction maps res

UV

respect the module

structures in the obvious sense, namely

res

UV

(fs) = res

UV

(f)res

UV

(s)

for all f ∈ A(U) and s ∈ M(U).

19