inverse limit
Let be a sequence of groups which arerelated by a chain of surjective
homomorphisms
such that
Definition 1.
The inverse limit of , denoted by
is the subset of formed byelements satisfying
Note: The inverse limit of can be checked to be a subgroupof the product
. See below for a more general definition.
Examples:
- 1.
Let be a prime. Let and. Define the connecting homomorphisms , for , tobe “reduction
modulo ” i.e.
which are obviously surjective homomorphisms. The inverse limit of is called the -adic integers anddenoted by
- 2.
Let be an elliptic curve defined over . Let be a prime and for any natural number
write for the-torsion group
, i.e.
In this case we define , and
The inverse limit of is called the Tate module of and denoted
The concept of inverse limit can be defined in far moregenerality. Let be a directed set and let be a category
. Let be a collection
of objects in the category and let
be a collection ofmorphisms satisfying:
- 1.
For all ,, the identitymorphism.
- 2.
For all such that , we have (composition of morphisms).
Definition 2.
The inverse limit of , denoted by
is defined to be the set of all such that for all
For a good example of this more general construction, see infiniteGalois 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 system![]() |
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 |