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:

F _II is the identity on has _I,
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 A_i :

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

This projective limit is provided naturally with projections on each A_i . Who more is, it checks a universal Propriété among the groups projecting itself on the A_i . 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 $\ mathbb {NR}$ and connected by the morphisms of reduction modulo p . An entirety p - adic is then a continuation $(a_n) _ {N \ Ge 1}$ such as $a_n \ in \ mathbb Z/p^n \ mathbb Z$ and that, if $n .$

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 X_i according to the morphisms F _ij, or by abuse language, limit according to I , even quite simply of projective limit of the X_i .

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

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