inverse limit

Let {Gi}i=0 be a sequencePlanetmathPlanetmath of groups which are related by a chain of surjectivePlanetmathPlanetmath homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath fi:GiGi-1 such that

Definition 1.

The inverse limitMathworldPlanetmathPlanetmath of (Gi,fi), denoted by

lim(Gi,fi), or limGi

is the subset of i=0Gi formed by elements satisfying


Note: The inverse limit of Gi can be checked to be a subgroupMathworldPlanetmathPlanetmath of the productPlanetmathPlanetmathPlanetmath i=0Gi. See below for a more general definition.


  1. 1.

    Let p be a prime. Let G0={0} and Gi=/pi. Define the connecting homomorphisms fi, for i2, to be “reductionPlanetmathPlanetmath modulo pi-1” i.e.


    which are obviously surjective homomorphisms. The inverse limit of (/pi,fi) is called the p-adic integers and denoted by

  2. 2.

    Let E be an elliptic curve defined over . Let p be a prime and for any natural numberMathworldPlanetmath n write E[n] for the n-torsion groupPlanetmathPlanetmath, i.e.


    In this case we define Gi=E[pi], and


    The inverse limit of (E[pi],fi) is called the Tate module of E and denoted


The conceptMathworldPlanetmath of inverse limit can be defined in far more generality. Let (S,) be a directed set and let 𝒞 be a categoryMathworldPlanetmath. Let {Gα}αS be a collectionMathworldPlanetmath of objects in the category 𝒞 and let


be a collection of morphisms satisfying:

  1. 1.

    For all αS, fα,α=IdGα, the identity morphism.

  2. 2.

    For all α,β,γS such that αβγ, we have fα,γ=fα,βfβ,γ (composition of morphisms).

Definition 2.

The inverse limit of ({Gα}αS,{fα,β}), denoted by

lim(Gα,fα,β), or limGα

is defined to be the set of all (gα)αSGα such that for all α,βS


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 systemMathworldPlanetmath
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