# Projective limit

In Mathematical, the concept of projective limit (opposite English limit) is used simultaneously to consider a whole family of objects, for example of the groups, bound between them by a family of morphisms, for example of the morphisms of groups.

The general framework for this concept is that of the categories.

## Concrete definition

To fix the ideas, one speaks initially about limit projective of groups.

One considers a family ( has I ) I I of groups, subscripted by a unit I ordered, and provided with a family of morphisms of groups F ij : has J has I for all I J (to notice the order: greater index the J towards smallest I , contrary to a inductive Limit) checking the conditions of compatibility:

1. F II is the identity on has I ,
2. F ik = F ij O F jk for all I J K .

The data ( I , has I , F ij ) is called projective system groups. The projective limit of this system is then defined like a sub-group of the produces direct Ai :

$\ varprojlim A_i = \ Big \ \left\{\left(a_i\right) \ in \ prod_ \left\{I \ in I\right\} A_i \; \ Big|\; \ forall I \ Leq J, \, \, a_i = f_ \left\{ij\right\} \left(a_j\right) \ Big \\right\}$

This projective limit is provided naturally with projections on each Ai . Who more is, it checks a universal Propriété among the groups projecting itself on the Ai . It is in fact possible to define the projective limit by this universal property.

Same construction can be carried out with units, rings, modules, algebras, instead of the groups.

### Example

The ring of the whole '' p '' - adic $\ mathbb Z_p$ is defined like the projective Limite of the rings $\ mathbb Z/p^n \ mathbb Z$, indexed by $\ mathbb \left\{NR\right\}$ and connected by the morphisms of reduction modulo p . An entirety p - adic is then a continuation $\left(a_n\right) _ \left\{N \ Ge 1\right\}$ such as $a_n \ in \ mathbb Z/p^n \ mathbb Z$ and that, if

## General standard

That is to say ( X I , F ij ) a projective system in a category C (the definition given above for the groups adapts to any category). The projective limit X is an object of the category C provided with arrows π I of X to values in X I checking the relations of compatibility π I = F ij O π J for all I J . Moreover, the data ( X , π I ) must be universal: for any other object Y provided with a family of arrows ψ I there exists a single arrow U : Y X such as the diagram:

that is to say commutative for all I J . The projective limit is noted: $X = \ varprojlim X_i$. One will speak about projective limit of the Xi according to the morphisms F ij , or by abuse language, limit according to I , even quite simply of projective limit of the Xi .

Contrary to the concrete case of the category of the groups, where the definition as sub-group of the direct product ensures the existence of a projective limit, the projective limit can not exist in a general category. On the other hand, unicity if existence is always assured: if X ′ is another projective limit, it exists a single isomorphism X ′ → X which commutates with projections.

## See too

 Random links: Gamekult.com | Ben Atiga | Pierre Andre Grobon | Johnny Griffin | University of Cergy-Pontoise | Périphéries_(album)