# Inverse limit

In mathematics, the **inverse limit** (also called the **projective limit** in the case of epimorphisms) is a construction that 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

We start with the definition of an **inverse** (or **projective** when the involved morphisms are epimorphisms) **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 on*A*_{i},*f*_{ik}=*f*_{ij}o*f*_{jk}for all*i*≤*j*≤*k*.

Then the pair ((*A*_{i})_{i∈I}, (*f*_{ij})_{i≤ j∈I}) 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})_{i∈I}, (*f*_{ij})_{i≤ j∈I}) as a particular subgroup of the direct product of the *A*_{i}'s:

The inverse limit, *A*, comes equipped with *natural projections* π_{i}: *A* → *A*_{i} which pick out the *i*th component of the direct product for each *i* in *I*. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the *A*_{i}'s are sets,^{[1]} semigroups,^{[1]} topological spaces,^{[1]} rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category.

### General definition

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} for all *i* ≤ *j*. The pair (*X*, π_{i}) must be universal in the sense that for any other such pair (*Y*, ψ_{i}) (i.e. ψ_{i}: *Y* → *X*_{i} with ψ_{i} = *f*_{ij} o ψ_{j} for all *i* ≤ *j*) there exists a unique morphism *u*: *Y* → *X* making all the "obvious" identities true; i.e., the diagram

must commute for all *i* ≤ *j*, for which it suffices to show that ψ_{i} = π_{i} o *u* for all *i*. The inverse limit is often denoted

with the inverse system (*X*_{i}, *f*_{ij}) being understood.

The inverse limit might not exist in a 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* if and only if *i* ≤ *j*. An inverse system is then just a contravariant functor *I* → *C*. And the inverse limit functor
is a covariant functor.

## Examples

- 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. - The ring of formal power series over a commutative ring
*R*can be thought of as the inverse limit of the rings , indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection. - 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).

- 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
- 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.

## Derived functors of the inverse limit

For an abelian category *C*, the inverse limit functor

is left exact. If *I* is ordered (not simply partially ordered) and countable, and *C* is the category **Ab** of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms *f*_{ij} that ensures the exactness of . Specifically, Eilenberg constructed a functor

(pronounced "lim one") such that if (*A*_{i}, *f*_{ij}), (*B*_{i}, *g*_{ij}), and (*C*_{i}, *h*_{ij}) are three projective systems of abelian groups, and

is a short exact sequence of inverse systems, then

is an exact sequence in **Ab**.

### Mittag-Leffler condition

If the ranges of the morphisms of an inverse system of abelian groups (*A*_{i}, *f*_{ij}) are *stationary*, that is, for every *k* there exists *j* ≥ *k* such that for all *i* ≥ *j* : one says that the system satisfies the **Mittag-Leffler condition**. This condition implies that

The following situations are examples where the Mittag-Leffler condition is satisfied:

- a system in which the morphisms
*f*_{ij}are surjective - a system of finite-dimensional vector spaces.

An example where this is non-zero is obtained by taking *I* to be the non-negative integers, letting *A*_{i} = *p*^{i}**Z**, *B*_{i} = **Z**, and *C*_{i} = *B*_{i} / *A*_{i} = **Z**/*p*^{i}**Z**. Then

where **Z**_{p} denotes the p-adic integers.

### Further results

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 *n*th right derived functor is denoted

In the case where *C* satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim^{1} on **Ab**^{I} to series of functors lim^{n} such that

It was thought for almost 40 years that Roos had proved (in *Sur les foncteurs dérivés de lim. Applications. *) that lim^{1} *A*_{i} = 0 for (*A*_{i}, *f*_{ij}) an inverse system with surjective transition morphisms and *I* the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences"). 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 (in "Derived functors of inverse limits revisited") that his result is correct if *C* has a set of generators (in addition to satisfying (AB3) and (AB4*)).

Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if *I* has cardinality (the *d*th infinite cardinal), then *R*^{n}lim is zero for all *n* ≥ *d* + 2. This applies to the *I*-indexed diagrams in the category of *R*-modules, with *R* a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which lim^{n}, on diagrams indexed by a countable set, is nonzero for n>1).

## Related concepts and generalizations

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.

## See also

## Notes

- ↑
^{1.0}^{1.1}^{1.2}John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.

## References

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- Section 3.5 of Template:Weibel IHA