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In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category.
## Formal definition

### Algebraic objects

_{i}: A → A_{i} which pick out the ith component of the direct product. The inverse limit and the natural projections satisfy a universal property described in the next section.### General definition

_{i}, f_{ij}) being understood. ## Examples

## Derived functors of the inverse limit

For an abelian category C, the inverse limit functor
_{ij} that ensures the exactness of $varprojlim$. Specifically, Eilenberg constructed a functor
_{i}, f_{ij}), (B_{i}, g_{ij}), and (C_{i}, h_{ij}) are three injective systems of abelian groups, and
_{i}, f_{ij}) is said to satisfy the Mittag-Leffler condition if for every k there exists j ≥ k such that for all i ≥ j
_{ij} are surjective satisfies the Mittag-Leffler condition. If (A_{i}, f_{ij}) satisfies the Mittag-Leffler condition, then
_{i} = p^{i}Z, B_{i} = Z, and C_{i} = B_{i} / A_{i} = Z/p^{i}Z. Then
_{p} denotes the p-adic integers.^{1} on Ab^{I} to series of functors lim^{n} such that
^{1} A_{i} = 0 for (A_{i}, f_{ij}) an inverse system with surjective transition morphisms and I the set of non-negative integers. However, in 2002, Amnon Neeman and Pierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim^{1} A_{i} ≠ 0. Roos has since shown that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).## Related concepts and generalizations

## References

We start with the definition of an inverse (or projective) system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (A_{i})_{i∈I} be a family of groups and suppose we have a family of homomorphisms f_{ij}: A_{j} → A_{i} for all i ≤ j (note the order) with the following properties:

- f
_{ii}is the identity in A_{i}, - f
_{ik}= f_{ij}o f_{jk}for all i ≤ j ≤ k.

Then the set of pairs (A_{i}, f_{ij}) is called an inverse system of groups and morphisms over I, and the morphisms f_{ij} are called the transition morphisms of the system.

We define the inverse limit of the inverse system (A_{i}, f_{ij}) as a particular subgroup of the direct product of the A_{i}'s:

- $varprojlim\; A\_i\; =\; Big\{(a\_i)\; in\; prod\_\{iin\; I\}A\_i\; ;Big|;\; a\_i\; =\; f\_\{ij\}(a\_j)\; mbox\{\; for\; all\; \}\; i\; leq\; jBig\}.$

This same construction may be carried out if the A_{i}'s are sets, rings, modules (over a fixed ring), algebras (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding category. The inverse limit will also belong to that category.

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (X_{i}, f_{ij}) be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_{i}: X → X_{i} (called projections) satisfying π_{i} = f_{ij} o π_{j}. The pair (X, π_{i}) must be universal in the sense that for any other such pair (Y, ψ_{i}) there exists a unique morphism u: Y → X making all the "obvious" identities true; i.e., the diagram

must commute for all i, j. The inverse limit is often denoted

- $X\; =\; varprojlim\; X\_i$

Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any other inverse limit X′ there exists a unique isomorphism X′ → X commuting with the projection maps.

We note that an inverse system in a category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. An inverse system is then just a contravariant functor I → C, and the inverse limit is a functor from the functor category C^{I} to the category C.

- The ring of p-adic integers is the inverse limit of the rings Z/p
^{n}Z (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here. - Pro-finite groups are defined as inverse limits of (discrete) finite groups.
- Let the index set I of an inverse system (X
_{i}, f_{ij}) have a greatest element m. Then the natural projection π_{m}: X → X_{m}is an isomorphism. - Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology.
- The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete, the limit space is totally disconnected. This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings).
- Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product.
- Let I consist of three elements i, j, and k with i ≤ j and i ≤ k (not directed). The inverse limit of any corresponding inverse system is the pullback.

- $varprojlim:C^Irightarrow\; C$

- $varprojlim\{\}^1:operatorname\{Ab\}^Irightarrowoperatorname\{Ab\}$

- $0rightarrow\; A\_irightarrow\; B\_irightarrow\; C\_irightarrow0$

- $0rightarrowvarprojlim\; A\_irightarrowvarprojlim\; B\_irightarrowvarprojlim\; C\_irightarrowvarprojlim\{\}^1A\_i$

- $f\_\{kj\}(A\_j)=f\_\{ki\}(A\_i).$

- $varprojlim\{\}^1A\_i=0.$

- $varprojlim\{\}^1A\_i=mathbf\{Z\}\_p/mathbf\{Z\}$

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C^{I}, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted

- $R^nvarprojlim:C^Irightarrow\; C.$

- $varprojlim\{\}^ncong\; R^nvarprojlim.$

Barry Mitchell has shown that if I has cardinality $aleph\_d$ (the dth infinite cardinal), then R^{n}lim is zero for all n ≥ d + 2.

The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.

- , section 3.5

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Last updated on Thursday October 02, 2008 at 19:04:51 PDT (GMT -0700)

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Last updated on Thursday October 02, 2008 at 19:04:51 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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