formal power series as inverse limits

formal power series as inverse limits

1

Motivation and Overview

The ring of

formal power series

can be described as an

inverse
limit

.

1

1

It is worth pointing out that, since we are
dealing with formal series, the

concept

of limit used here has
nothing to do with convergence but is purely

algebraic

.

The
fundamental idea behind this approach is that of

truncation

—
given a formal power series

a

0

+

a

1

⁢

t

+

a

2

⁢

t

2

+

a

3

⁢

t

3

+

⋯

and an integer

n

≥

0

, we can truncate the series to
order

n

to obtain

a

0

+

a

1

⁢

t

+

⋯

+

a

n

⁢

t

n

+

O

⁢

(

t

n

+

1

)

.

2

2

Here, the symbol “

O

⁢

(

t

n

+

1

)

” is not
used in the sense of

Landau notation

but merely as an indicator
that the power series has been truncated to order

n

.

(Indeed, we must do this in practical computation since it is
only possible to write down a finite number of terms at a time;
thus the approach taken here has the advantage of being close
to actual practice.) Furthermore, this procedure of truncation
commutes with ring

operations

— given two formal power series,
the truncation of their sum is the sum of their truncations and
the truncation of their

product

is the product of their
truncations. Thus, for every integer

n

the set of power
series truncated to order

n

forms a ring and truncation is a

morphism

from the ring of formal power series to this ring.

To obtain our

definition

, we will proceed in the opposite
direction. We will begin by defining rings of truncated
power series and exhibiting truncation morphisms between
different truncations. Then we will show that these rings
and morphisms form an

inverse system

which has a limit which
we will take as the definition of the ring of formal power
series. Finally, we will

complete

the circle by demonstrating
that the object so constructed is

isomorphic

with the usual
definition for ring of formal power series.

2

Formal Development

In this

section

, we will carry out the develpment outlined
above in

rigorous

detail. We begin by formalizing this
notion of truncation.

Theorem 1

.

Let

A

be a

commutative ring

and let

n

be a positive
integer. Then

A

⁢

[

[

x

]

]

/

⟨

x

n

⟩

is isomorphic
to

A

⁢

[

x

]

/

⟨

x

n

⟩

.

Proof.

We may identify

A

⁢

[

x

]

with the

subring

of

A

⁢

[

[

x

]

]

consisting
of series which have all but a finite number of coeficients
equal to zero. Consider an

element

f

=

∑

k

=

0

∞

c

n

⁢

x

n

of

A

⁢

[

[

x

]

]

. We may write

f

=

∑

k

=

0

n

-

1

c

n

⁢

x

k

+

x

n

⁢

∑

k

=

0

∞

c

k

+

n

⁢

x

k

. Thus, every element of

A

⁢

[

[

x

]

]

is

equivalent

to an element of the subring

A

⁢

[

x

]

modulo

x

k

. Hence, if follows rather immediately from the
definition of

quotient ring

that

A

⁢

[

[

x

]

]

⁢

⟨

x

n

⟩

is isomorphic to

A

⁢

[

x

]

/

⟨

x

n

⟩

.
∎

Let us call the

isomorphism

between

A

⁢

[

[

x

]

]

/

⟨

x

n

⟩

and

A

⁢

[

x

]

/

⟨

x

n

⟩

which is described above

I

n

. We now define a few more morphisms.

Definition 1

.

Suppose

m

,

n

are integers satifying the inequalities

m

>

n

≥

0

. Then define the morphisms

t

m

⁢

n

,

T

m

⁢

n

,

p

n

,

P

n

as follows:

•

Define

t

n

⁢

m

:

A

⁢

[

x

]

/

⟨

x

m

⟩

→

A

⁢

[

x

]

/

⟨

x

n

⟩

as the map which sends each

equivalence class

a

modulo

x

n

to the unique equivalence
class

b

modulo

x

m

such that

a

⊂

b

.

•

Define

T

n

⁢

m

:

A

⁢

[

[

x

]

]

/

⟨

x

m

⟩

→

A

⁢

[

[

x

]

]

/

⟨

x

n

+

1

⟩

as the map which sends each
equivalence class

a

modulo

x

n

to the unique equivalence
class

b

modulo

x

m

such that

a

⊂

b

.

•

For every integer

n

>

0

, let

Q

n

be the quotient
map from

A

⁢

[

[

x

]

]

to

A

⁢

[

[

x

]

]

/

⟨

x

n

⟩

.

•

For every integer

n

>

0

, let

q

n

be the

quotient

map from

A

⁢

[

x

]

to

A

⁢

[

x

]

/

⟨

x

n

⟩

.

These morphisms commute with each other in ways which are
described by the next

theorem

:

Theorem 2

.

Suppose

m

,

n

,

k

are integers satifying the inequalities

m

>

n

>

k

≥

0

. Then we have the following

relations

:

1.

t

n

⁢

k

∘

t

m

⁢

n

=

t

m

⁢

k

2.

I

n

∘

T

m

⁢

n

=

t

m

⁢

n

∘

I

m

3.

T

n

⁢

k

∘

T

m

⁢

n

=

T

m

⁢

k

4.

t

m

⁢

n

∘

q

m

=

q

n

5.

T

m

⁢

n

∘

Q

m

=

Q

n

[More to come]

Title

formal power series as inverse limits

Canonical

name

FormalPowerSeriesAsInverseLimits

Date of creation

2013-03-22 18:22:41

Last modified on

2013-03-22 18:22:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Result

Classification

msc 13F25

Classification

msc 13B35

Classification

msc 13J05

Classification

msc 13H05

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