Inverse limit

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Introduction

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.

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Formal definition

The formal definition consists of the following steps:

Definition of the inverse system that creates mappings between groups
the mappings are homomorphisms between the groups

Family of Groups

We start with the definition of an inverse system (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.

Family of Group Homomorphisms

Suppose we have a family of homomorphisms $f_{ij}:A_{j}\to A_{i}$ for all $i\leq j$ (note the order) with the following properties:

$f_{ii}$ is the identity on $A_{i}$ ,
$f_{ik}=f_{ij}\circ f_{jk}\quad {\text{ for all }}\quad i\leq j\leq k.$

Then the pair $((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})$ is called an inverse system of groups and morphisms over $I$ , and the morphisms $f_{ij}$ are called the transition morphisms of the system.

Inverse System and Inverse Limit

We define the inverse limit of the inverse system $((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})$ as a particular subgroup of the direct product of the $A_{i}$ 's:

A=\varprojlim _{i\in I}{A_{i}}=\left\{\left.{\vec {a}}\in \prod _{i\in I}A_{i}\;\right|\;a_{i}=f_{ij}(a_{j}){\text{ for all }}i\leq j{\text{ in }}I\right\}.

Remark

The inverse limit $A$ comes equipped with natural projections $\pi \setminus \!\!\!\setminus \pi _{i}$ >: $A$ → $A_{i}$ which pick out the ith 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 ${\textstyle (X_{i},f_{ij})}$ 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 $\pi$ _i: X → X_i (called projections) satisfying $\pi$ _i = $f_{ij}$ ∘ $\pi$ _j for all i ≤ j.

Diagram

The pair (X, $\pi$ _i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u: Y → X such that the diagram

commutes for all i ≤ j. The inverse limit is often denoted

X=\varprojlim X_{i}

with the inverse system ${\textstyle (X_{i},f_{ij})}$ being understood.

Existence and Uniqueness of Inverse Systems

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 $X':\to$ commuting with the projection maps.

Category C and partially ordered sets

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 $C^{I^{\mathrm {op} }}$ 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 $C^{I^{\mathrm {op} }}.$ The direct limit, if it exists, is defined as a right adjoint of this trivial functor.

Examples

The following list provides examples of inverse limits

Example 1 - p-adic Integers

The ring of p-adic integers is the inverse limit of the rings $\mathbb {Z} /p^{n}\mathbb {Z}$ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers $(n_{1},n_{2},\dots )$ such that each element of the sequence "projects" down to the previous ones, namely, that $n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}$ whenever $i<j.$ The natural topology on the p-adic integers is the one implied here, namely the product topology with cylinder sets as the open sets.

Example 2 - p-adic Solenoids

The ring of p-adic solenoids is the inverse limit of the rings $\mathbb {R} /p^{n}\mathbb {Z}$ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers $(n_{1},n_{2},\dots )$ such that each element of the sequence "projects" down to the previous ones, namely, that $n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}$ whenever $i<j.$

Example 3 - Factor Groups

Let p be a prime number. Consider the direct system composed of the factor groups $\mathbb {Z} /p^{n}\mathbb {Z}$ and the homomorphisms $\mathbb {Z} /p^{n}\mathbb {Z} \rightarrow \mathbb {Z} /p^{n+1}\mathbb {Z}$ induced by multiplication by $p$ . The inverse limit of this system is the circle group $\mathbb {T} =\mathbb {R} /\mathbb {Z}$ , expressed in a positional number system of base $p$ . This is similar to the construction of the real numbers by Cauchy sequences, using elements of the Prüfer group instead of the rational numbers or base $p$ decimal fractions.

Example 4 - Formal Power Series

The ring $R[t]$ of formal power series over a commutative ring R can be thought of as the inverse limit of the rings $\textstyle R[t]/t^{n}R[t]$ , indexed by the natural numbers as usually ordered, with the morphisms from $\textstyle R[t]/t^{n+j}R[t]$ to $\textstyle R[t]/t^{n}R[t]$ given by the natural projection.

Example 5 - Pro-finite Groups

Pro-finite groups are defined as inverse limits of (discrete) finite groups.

Example 6 - Natural Projections

Let the index set I of an inverse system (X_i, $f_{ij}$ ) have a greatest element m. Then the natural projection $\pi$ _m: X → X_m is an isomorphism.

Example 7 - Inverse system and inverse limits - Category Sets

In the category of sets, every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of Kőnig's lemma in graph theory and may be proved with Tychonoff's theorem, viewing the finite sets as compact discrete spaces, and then applying the finite intersection property characterization of compactness.

Example 8 - Topological Spaces

In the category of topological spaces, every inverse system has an inverse limit. It is constructed by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology.

Example 9 - Set of Strings in Computer Science

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).

Derived functors of the inverse limit

For an abelian category C, the inverse limit functor

\varprojlim :C^{I}\rightarrow C

is left exact.

Ordered and Countable

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 $\varprojlim$ . Specifically, Eilenberg constructed a functor

\varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab}

(pronounced "lim one") such that if (A_i, f_ij), (B_i, g_ij), and (C_i, h_ij) are three inverse systems of abelian groups, and

0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0

is a short exact sequence of inverse systems, then

0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{i}

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 : $f_{kj}(A_{j})=f_{ki}(A_{i})$ one says that the system satisfies the Mittag-Leffler condition.

Origine of the Name

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.

Examples for Mittag-Leffler condition

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 or finite abelian groups or modules of finite length or Artinian modules.

An example where $\varprojlim {}^{1}$ is non-zero is obtained by taking I to be the non-negative integers, letting A_i = pⁱZ, B_i = Z, and C_i = B_i / A_i = Z/pⁱZ. Then

\varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z}

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 nth right derived functor is denoted

R^{n}\varprojlim :C^{I}\rightarrow C.

In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim¹ on Ab^I to series of functors limⁿ such that

\varprojlim {}^{n}\cong R^{n}\varprojlim .

It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim¹ 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 lim¹ 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 $\aleph _{d}$ (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).

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 a class of limits, while direct limits are a class of colimits.

Learning Tasks / Activities

Explain how the ordered sets in $I$ is linked to properties of the group homomorphisms.

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

Bourbaki, Nicolas (1989), Algebra I, Springer, ISBN 978-3-540-64243-5, OCLC 40551484 {{citation}}: Check |authorlink= value (help)
Bourbaki, Nicolas (1989), General topology: Chapters 1-4, Springer, ISBN 978-3-540-64241-1, OCLC 40551485 {{citation}}: Check |authorlink= value (help)
Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0-387-98403-8 {{citation}}: Check |authorlink= value (help)
Mitchell, Barry (1972), "Rings with several objects", Advances in Mathematics - MR 0294454, 8: 1–161, doi:10.1016/0001-8708(72)90002-3 {{citation}}: Check |author-link= value (help)
Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)", Inventiones Mathematicae MR: 1906154, 148 (2): 397–420, doi:10.1007/s002220100197 {{citation}}: Check |author-link= value (help)
Roos, Jan-Erik (1961), "Sur les foncteurs dérivés de lim. Applications", C. R. Acad. Sci. Paris MR: 0132091, 252: 3702–3704 {{citation}}: Check |author-link= value (help)
Roos, Jan-Erik (2006), "Derived functors of inverse limits revisited", J. London Math. Soc., Series 2 - MR 2197371, 73 (1): 65–83, doi:10.1112/S0024610705022416 {{citation}}: Check |author-link= value (help)

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