inverse limit

inverse limit

Let

{

G

i

}

i

=

0

∞

be a

sequence

of groups which are
related by a chain of

surjective

homomorphisms

f

i

:

G

i

→

G

i

-

1

such that

\xymatrix

⁢

G

0

⁢

&

⁢

G

1

⁢

\ar

⁢

@

-

>

[

l

]

f

1

⁢

&

⁢

G

2

⁢

\ar

⁢

@

-

>

[

l

]

f

2

⁢

&

⁢

G

3

⁢

\ar

⁢

@

-

>

[

l

]

f

3

⁢

&

⁢

…

⁢

\ar

⁢

@

-

>

[

l

]

f

4

Definition 1

.

The

inverse limit

of

(

G

i

,

f

i

)

, denoted by

lim

←

⁡

(

G

i

,

f

i

)

,

or

lim

←

⁡

G

i

is the subset of

∏

i

=

0

∞

G

i

formed by

elements

satisfying

(

g

0

,

g

1

,

g

2

,

g

3

,

…

)

,

𝑤𝑖𝑡ℎ

g

i

∈

G

i

,

f

i

⁢

(

g

i

)

=

g

i

-

1

Note: The inverse limit of

G

i

can be checked to be a

subgroup

of the

product

∏

i

=

0

∞

G

i

. See below for a more general definition.

Examples

:

1.

Let

p

∈

ℕ

be a prime. Let

G

0

=

{

0

}

and

G

i

=

ℤ

/

p

i

⁢

ℤ

. Define the

connecting homomorphisms

f

i

, for

i

≥

2

, to
be “

reduction

modulo

p

i

-

1

” i.e.

f

i

:

ℤ

/

p

i

⁢

ℤ

→

ℤ

/

p

i

-

1

⁢

ℤ

f

i

⁢

(

x

⁢

mod

⁡

p

i

)

=

x

⁢

mod

⁡

p

i

-

1

which are obviously surjective homomorphisms. The inverse limit of

(

ℤ

/

p

i

⁢

ℤ

,

f

i

)

is called the

p

-adic integers and
denoted by

ℤ

p

=

lim

←

⁡

ℤ

/

p

i

⁢

ℤ

2.

Let

E

be an elliptic curve defined over

ℂ

. Let

p

be a prime and for any

natural number

n

write

E

⁢

[

n

]

for the

n

-

torsion group

, i.e.

E

⁢

[

n

]

=

{

Q

∈

E

∣

n

⋅

Q

=

O

}

In this case we define

G

i

=

E

⁢

[

p

i

]

, and

f

i

:

E

⁢

[

p

i

]

→

E

⁢

[

p

i

-

1

]

,

f

i

⁢

(

Q

)

=

p

⋅

Q

The inverse limit of

(

E

⁢

[

p

i

]

,

f

i

)

is called the Tate module of

E

and denoted

T

p

⁢

(

E

)

=

lim

←

⁡

E

⁢

[

p

i

]

The

concept

of inverse limit can be defined in far more
generality. Let

(

S

,

≤

)

be a

directed set

and let

𝒞

be a

category

. Let

{

G

α

}

α

∈

S

be a

collection

of objects in the category

𝒞

and let

{

f

α

,

β

:

G

β

→

G

α

∣

α

,

β

∈

S

,

α

≤

β

}

be a collection of

morphisms

satisfying:

1.

For all

α

∈

S

,

f

α

,

α

=

Id

G

α

, the identity
morphism.

2.

For all

α

,

β

,

γ

∈

S

such that

α

≤

β

≤

γ

, we have

f

α

,

γ

=

f

α

,

β

∘

f

β

,

γ

(

composition

of morphisms).

Definition 2

.

The

inverse limit

of

(

{

G

α

}

α

∈

S

,

{

f

α

,

β

}

)

, denoted by

lim

←

⁡

(

G

α

,

f

α

,

β

)

,

or

lim

←

⁡

G

α

is defined to be the set of all

(

g

α

)

∈

∏

α

∈

S

G

α

such that for all

α

,

β

∈

S

α

≤

β

⇒

f

α

,

β

⁢

(

g

β

)

=

g

α

For a good example of this more general construction, see

infinite
Galois theory

.

Title

inverse limit

Canonical name

InverseLimit

Date of creation

2013-03-22 13:54:20

Last modified on

2013-03-22 13:54:20

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Definition

Classification

msc 20F22

Synonym

inverse system

Synonym

projective limit

Related topic

PAdicIntegers

Related topic

GaloisRepresentation

Related topic

InfiniteGaloisTheory

Related topic

ProfiniteGroup

Related topic

CategoryAssociatedToAPartialOrder

Related topic

DirectLimit

Related topic

CohomologyOfSmallCategories

Defines

inverse limit

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