In mathematics, the inverse limit (also called the projective limit) 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, and they are a special case of the concept of a limit in category theory.
We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family of groups and suppose we have a family of homomorphisms for all (note the order) with the following properties:
Then the pair is called an inverse system of groups and morphisms over , and the morphisms are called the transition morphisms of the system.
We define the inverse limit of the inverse system as a particular subgroup of the direct product of the 's:
The inverse limit comes equipped with natural projectionsπ \\ πi: → which pick out the ith component of the direct product for each in . 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 's are sets, semigroups, topological spaces, 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.
The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let 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: X → Xi (called projections) satisfying πi = ∘ πj for all i ≤ j. The pair (X, πi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u: Y → X such that the diagram
commutes for all i ≤ j. The inverse limit is often denoted
with the inverse system being understood.
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits X and X' of an inverse system, there exists a unique isomorphism X′ → X commuting with the projection maps.
Inverse systems and inverse limits in a category C admit 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. Let be the category of these functors (with natural transformations as morphisms). An object X of C can be considered a trivial inverse system, where all objects are equal to X and all arrow are the identity of X. This defines a "trivial functor" from C to The direct limit, if it exists, is defined as a right adjoint of this trivial functor.
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 fij that ensures the exactness of . Specifically, Eilenberg constructed a functor
(pronounced "lim one") such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three inverse systems of abelian groups, and
is a short exact sequence of inverse systems, then
is an exact sequence in Ab.
If the ranges of the morphisms of an inverse system of abelian groups (Ai, fij) 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.
The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof of Mittag-Leffler's theorem.
The following situations are examples where the Mittag-Leffler condition is satisfied:
An example where 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
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
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
It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim1Ai = 0 for (Ai, fij) 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 lim1Ai ≠ 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 dth infinite cardinal), then Rnlim 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 limn, on diagrams indexed by a countable set, is nonzero for n > 1).
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