Definitions

Inverse_limit

Inverse limit

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

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 (Ai)iI be a family of groups and suppose we have a family of homomorphisms fij: AjAi for all ij (note the order) with the following properties:

  1. fii is the identity in Ai,
  2. fik = fij o fjk for all ijk.

Then the set of pairs (Ai, fij) is called an inverse system of groups and morphisms over I, and the morphisms fij are called the transition morphisms of the system.

We define the inverse limit of the inverse system (Ai, fij) as a particular subgroup of the direct product of the Ai'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}.
The inverse limit, A, comes equipped with natural projections πi: AAi 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.

This same construction may be carried out if the Ai'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.

General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (Xi, fij) 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 πi: XXi (called projections) satisfying πi = fij o πj. The pair (X, πi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u: YX 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
with the inverse system (Xi, fij) being understood.

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 ij iff ij. An inverse system is then just a contravariant functor IC, and the inverse limit is a functor from the functor category CI to the category C.

Examples

  • The ring of p-adic integers is the inverse limit of the rings Z/pnZ (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 (Xi, fij) have a greatest element m. Then the natural projection πm: XXm 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.
  • 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 ij and ik (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
varprojlim:C^Irightarrow C
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 fij that ensures the exactness of varprojlim. Specifically, Eilenberg constructed a functor
varprojlim{}^1:operatorname{Ab}^Irightarrowoperatorname{Ab}
(pronounced "lim one") such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three injective systems of abelian groups, and
0rightarrow A_irightarrow B_irightarrow C_irightarrow0
is a short exact sequence of inverse systems, then
0rightarrowvarprojlim A_irightarrowvarprojlim B_irightarrowvarprojlim C_irightarrowvarprojlim{}^1A_i
is an exact sequence in Ab. An inverse system of abelian groups (Ai, fij) is said to satisfy the Mittag-Leffler condition if for every k there exists jk such that for all ij
f_{kj}(A_j)=f_{ki}(A_i).
For example, a system in which the morphisms fij are surjective satisfies the Mittag-Leffler condition. If (Ai, fij) satisfies the Mittag-Leffler condition, then
varprojlim{}^1A_i=0.
An example where this is non-zero is obtained by taking I to be the non-negative integers, letting Ai = piZ, Bi = Z, and Ci = Bi / Ai = Z/piZ. Then
varprojlim{}^1A_i=mathbf{Z}_p/mathbf{Z}
where Zp denotes the p-adic integers.

More generally, if C is an arbitrary abelian category that has enough injectives, then so does CI, 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.
In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim1 on AbI to series of functors limn such that
varprojlim{}^ncong R^nvarprojlim.
It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim1 Ai = 0 for (Ai, fij) 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 lim1 Ai ≠ 0. Roos has since shown that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).

Barry Mitchell has shown that if I has cardinality aleph_d (the dth infinite cardinal), then Rnlim is zero for all nd + 2.

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.

References

  • , section 3.5

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