formal power series as inverse limits
1 Motivation and Overview
The ring of formal power series can be described as an inverse
limit.11It 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 and an integer , we can truncate the series to
order to obtain .22Here, the symbol “” is not
used in the sense of Landau notation but merely as an indicator
that the power series has been truncated to order .
(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 the set of power
series truncated to order 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 be a commutative ring and let be a positive integer. Then is isomorphic to .
Proof.
We may identify with the subring of consisting
of series which have all but a finite number of coeficients
equal to zero. Consider an element of . We may write . Thus, every element of
is equivalent to an element of the subring
modulo . Hence, if follows rather immediately from the
definition of quotient ring
that
is isomorphic to .
∎
Let us call the isomorphism between
and
which is described above . We now define a few more morphisms.
Definition 1.
Suppose are integers satifying the inequalities . Then define the morphisms as follows:
-
•
Define as the map which sends each equivalence class
modulo to the unique equivalence class modulo such that .
-
•
Define as the map which sends each equivalence class modulo to the unique equivalence class modulo such that .
-
•
For every integer , let be the quotient map from to .
-
•
For every integer , let be the quotient
map from to .
These morphisms commute with each other in ways which are
described by the next theorem:
Theorem 2.
Suppose are integers satifying the inequalities
. Then we have the following relations:
-
1.
-
2.
-
3.
-
4.
-
5.
[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 |