formal power series as inverse limits

1 Motivation and Overview

The ring of formal power series can be described as an inverse limitMathworldPlanetmathPlanetmath.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 a0+a1t+a2t2+a3t3+ and an integer n0, we can truncate the series to order n to obtain a0+a1t++antn+O(tn+1).22Here, the symbol “O(tn+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 operationsMathworldPlanetmath — given two formal power series, the truncation of their sum is the sum of their truncations and the truncation of their productMathworldPlanetmathPlanetmathPlanetmath 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 morphismMathworldPlanetmathPlanetmath 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 systemMathworldPlanetmath which has a limit which we will take as the definition of the ring of formal power series. Finally, we will completePlanetmathPlanetmathPlanetmathPlanetmath the circle by demonstrating that the object so constructed is isomorphicPlanetmathPlanetmathPlanetmath with the usual definition for ring of formal power series.

2 Formal Development

In this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, 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]]/xn is isomorphic to A[x]/xn.


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=0cnxn of A[[x]]. We may write f=k=0n-1cnxk+xnk=0ck+nxk. Thus, every element of A[[x]] is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to an element of the subring A[x] modulo xk. Hence, if follows rather immediately from the definition of quotient ringMathworldPlanetmath that A[[x]]xn is isomorphic to A[x]/xn. ∎

Let us call the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath between A[[x]]/xn and A[x]/xn which is described above In. We now define a few more morphisms.

Definition 1.

Suppose m,n are integers satifying the inequalities m>n0. Then define the morphisms tmn,Tmn,pn,Pn as follows:

  • Define tnm:A[x]/xmA[x]/xn as the map which sends each equivalence classMathworldPlanetmath a modulo xn to the unique equivalence class b modulo xm such that ab.

  • Define Tnm:A[[x]]/xmA[[x]]/xn+1 as the map which sends each equivalence class a modulo xn to the unique equivalence class b modulo xm such that ab.

  • For every integer n>0, let Qn be the quotient map from A[[x]] to A[[x]]/xn.

  • For every integer n>0, let qn be the quotientPlanetmathPlanetmath map from A[x] to A[x]/xn.

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

Theorem 2.

Suppose m,n,k are integers satifying the inequalities m>n>k0. Then we have the following relationsMathworldPlanetmathPlanetmathPlanetmath:

  1. 1.


  2. 2.


  3. 3.


  4. 4.


  5. 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