Inverse limit

In mathematics, the inverse limit (also called the projective limit or 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.

Formal definition

Algebraic objects

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 and suppose we have a family of homomorphisms f_ij: A_j → A_i for all i ≤ j (note the order), called bonding maps, 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:

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

The inverse limit, A, comes equipped with natural projections π_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 (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 such that the diagram

commutes 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

X=\varprojlim X_{i}

with the inverse system (X_i, f_ij) being understood.

In some categories, the inverse limit does not exist. 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 $\varprojlim :C^{I^{op}}\rightarrow C$ is a covariant functor.

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". That is, one considers sequences of integers $(n_{0},n_{1},\cdots )$ such that each element of the sequence "projects" down to the previous ones, namely, that $n_{i}\equiv n_{j}{\mbox{ mod }}p^{i+1}$ 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.
The ring $\textstyle R[[t]]$ 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.
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.

Derived functors of the inverse limit

For an abelian category C, the inverse limit functor

\varprojlim :C^{I}\rightarrow 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 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 projective 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. This condition implies that $\varprojlim {}^{1}A_{i}=0.$

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:

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 limits, while direct limits are colimits.

Notes

1 2 3 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
Bourbaki, Nicolas (1989), General topology: Chapters 1-4, Springer, ISBN 978-3-540-64241-1, OCLC 40551485
Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0-387-98403-8
Mitchell, Barry (1972), "Rings with several objects", Advances in Mathematics, 8: 1–161, doi:10.1016/0001-8708(72)90002-3, MR 0294454
Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)", Inventiones Mathematicae, 148 (2): 397–420, doi:10.1007/s002220100197, MR 1906154
Roos, Jan-Erik (1961), "Sur les foncteurs dérivés de lim. Applications", C. R. Acad. Sci. Paris, 252: 3702–3704, MR 0132091
Roos, Jan-Erik (2006), "Derived functors of inverse limits revisited", J. London Math. Soc. (2), 73 (1): 65–83, doi:10.1112/S0024610705022416, MR 2197371
Section 3.5 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259. MR 1269324.

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