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.
Concrete definitionTo 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 Ai :
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.
The ring of the whole '' p '' - adic is defined like the projective Limite of the rings , indexed by and connected by the morphisms of reduction modulo p . An entirety p - adic is then a continuation such as and that, if