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单词 InverseLimit
释义

inverse limit


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

\\xymatrixG0&G1\\ar@->[l]f1&G2\\ar@->[l]f2&G3\\ar@->[l]f3&\\ar@->[l]f4
Definition 1.

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

lim(Gi,fi), or limGi

is the subset of i=0Gi formed byelements satisfying

(g0,g1,g2,g3,),𝑤𝑖𝑡ℎgiGi,fi(gi)=gi-1

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

Examples:

  1. 1.

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

    fi:/pi/pi-1
    fi(xmodpi)=xmodpi-1

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

    p=lim/pi
  2. 2.

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

    E[n]={QEnQ=O}

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

    fi:E[pi]E[pi-1],fi(Q)=pQ

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

    Tp(E)=limE[pi]

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

{fα,β:GβGαα,βS,αβ}

be a collection ofmorphisms satisfying:

  1. 1.

    For all αS,fα,α=IdGα, the identitymorphism.

  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

αβfα,β(gβ)=gα

For a good example of this more general construction, see infiniteGalois theory.

Titleinverse limit
Canonical nameInverseLimit
Date of creation2013-03-22 13:54:20
Last modified on2013-03-22 13:54:20
Owneralozano (2414)
Last modified byalozano (2414)
Numerical id10
Authoralozano (2414)
Entry typeDefinition
Classificationmsc 20F22
Synonyminverse systemMathworldPlanetmath
Synonymprojective limit
Related topicPAdicIntegers
Related topicGaloisRepresentation
Related topicInfiniteGaloisTheory
Related topicProfiniteGroup
Related topicCategoryAssociatedToAPartialOrder
Related topicDirectLimit
Related topicCohomologyOfSmallCategories
Definesinverse limit
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