sheaf of sections

sheaf of sections

0.1

Presheaf Definition

Consider a rank

r

vector bundle

E

→

M

, whose typical fibre is defined with respect to a field

k

. Let

{

U

α

}

constitute a cover
for

M

. Then,

sections

of the bundle over some

U

⊂

M

are defined
as

continuous functions

U

→

E

, which commute with the

natural projection

map

π

:

E

→

M

;

π

∘

s

=

i

⁢

d

M

. Denote the space of sections of
the bundle over U to be

Γ

⁢

(

U

,

E

)

. The space of sections is a

vector
space

over the field

k

by defining

addition

and

scalar multiplication

pointwise

: for

s

,

t

∈

Γ

⁢

(

U

,

E

)

,

p

∈

U

and

a

∈

k

(

s

+

t

)

⁢

(

p

)

≡

s

⁢

(

p

)

+

t

⁢

(

p

)

(

a

⋅

s

)

⁢

(

p

)

≡

a

⋅

s

⁢

(

p

)

.

Then, this forms a

presheaf

ℰ

, a functor from

(

(

top

M

)

)

to the category of vector spaces, with

restriction

maps the natural

restriction of functions

.

0.2

Sheaf Axioms

It is easy to see that it

satisfies

the sheaf
axioms: for

U

open and

{

V

i

}

a cover of

U

,

1.

if

s

∈

ℰ

⁢

(

U

)

and

s

|

V

i

=

0

for all

i

, then

s

=

0

.

2.

if

s

i

∈

ℰ

⁢

(

V

i

)

for all

i

, such that for each

i

,

j

with

V

i

∩

V

j

≠

∅

,

s

i

|

V

i

∩

V

j

=

s

j

|

V

i

∩

V

j

, then there is an

s

∈

ℰ

⁢

(

U

)

with

s

|

V

i

=

s

i

for
all

i

.

The first follows from the fact that for any

U

, there is always at least
one element of

ℰ

⁢

(

U

)

, the

zero section

, and that the

transition
functions

of the bundle are

linear maps

. The second follows by the construction of the bundle.

1

Sheafification

We may also see the vector bundle by applying associated sheaf construction
to the presheaf

U

↦

Γ

⁢

(

U

,

E

)

. First though, we show that the
stalk of the sheaf

ℰ

at a point is

isomorphic

to the fibre of the
bundle

E

at the point. Let

[

s

,

U

]

be a germ at

p

∈

M

(

p

∈

U

⊂

M

)

, and define a map

ψ

:

ℰ

p

→

E

p

by

ψ

:

[

s

,

U

]

↦

s

p

.

First, we show that the map is a vector space homomorphism. Consider two
germs

[

s

,

U

]

and

[

t

,

V

]

in

ℰ

p

. These map to

s

p

and

t

p

respectively. We add the germs by finding an open set

W

∈

U

∩

V

and
adding the restrictions of the sections;

[

s

,

U

]

+

[

t

,

V

]

≡

[

s

|

W

+

t

|

W

,

W

]

.

Of course,

p

∈

W

, so we have

ψ

⁢

(

s

|

W

+

t

|

W

)

=

s

p

+

t

p

, since the
restriction maps are simply restriction of functions.
Now, it is easy to show that

ψ

is

injective

.
Assume

ψ

⁢

(

[

t

,

V

]

)

=

ψ

⁢

(

[

s

,

U

]

)

=

s

p

. Then

ψ

⁢

(

[

t

,

V

]

)

-

ψ

⁢

(

[

s

,

U

]

)

=

s

p

-

s

p

ψ

⁢

(

[

t

,

V

]

-

[

s

,

U

]

)

=

0

=

[

s

,

U

]

Now, we show that

ψ

is

surjective

. For

s

p

∈

E

p

, let

U

⊂

M

open be isomorphic to some subset

U

ℝ

of

ℝ

m

. Then,

Γ

⁢

(

U

,

E

)

is the
set of continuous maps

U

→

V

E

, where

V

E

is the typical fibre of

E

;

Γ

⁢

(

U

,

E

)

=

⊕

i

=

1

r

𝒞

U

ℝ

∞

.

Then let

[

s

,

U

]

be the

constant function

s

:

U

ℝ

↦

s

x

, and we
have constructed an

isomorphism

ψ

between

ℰ

p

and

E

p

.

To construct the Étalé space, take the

disjoint union

of stalks,

Spé

⁢

(

ℰ

)

=

∐

p

∈

M

ℰ

p

, and endow it with
the following

topology

: the open sets shall be of the form

U

s

=

{

s

p

|

s

∈

Γ

⁢

(

U

,

ℰ

)

,

p

∈

U

⊂

M

}

,

collection

of germs of sections at points in

U

⊂

M

.

Then, the associated sheaf to

ℰ

is the presheaf which assigns
continuous maps

Γ

⁢

(

U

,

Spé

⁢

(

ℰ

)

)

to each open

U

. These are
maps where the

preimage

of

U

s

is open. Clearly, this implies that

Γ

⁢

(

U

,

E

)

⊂

Γ

⁢

(

U

,

Spé

⁢

(

ℰ

)

)

. To go the other way, note
that open sets of

Spé

⁢

(

ℰ

)

are the images of continuous maps

U

→

E

. An open subset of

Spé

⁢

(

ℰ

)

may be written as a union of

U

t

;

U

t

⁢

s

≡

{

t

p

,

s

p

|

p

∈

U

}

. Then, by single-valuedness of maps,
a continuous map

U

→

Spé

⁢

(

ℰ

)

must map to

U

t

for some

t

∈

Γ

⁢

(

U

,

E

)

, so we have

Γ

⁢

(

U

,

E

)

⊃

Γ

⁢

(

U

,

Spé

⁢

(

ℰ

)

)

.

Title

sheaf of sections

Canonical name

SheafOfSections

Date of creation

2013-03-22 15:46:36

Last modified on

2013-03-22 15:46:36

Owner

guffin (12505)

Last modified by

guffin (12505)

Numerical id

7

Author

guffin (12505)

Entry type

Definition

Classification

msc 55R25

Related topic

VectorBundle

Defines

Sheaf of Sections

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