Inverse limit

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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[edit]

Algebraic objects[edit]

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 (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 on Ai,
  2. fik = fij o fjk for all ijk.

Then the pair ((Ai)iI, (fij)ijI) 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)iI, (fij)ijI) as a particular subgroup of the direct product of the Ai's:

\varprojlim_{i\in I} A_i = \Big\{\vec a \in \prod_{i\in I}A_i \;\Big|\; a_i = f_{ij}(a_j) \mbox{ for all } i \leq j \mbox{ in } I\Big\}.

The inverse limit, A, comes equipped with natural projections πi: AAi 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 Ai's are sets,[1] semigroups,[1] topological spaces,[1] 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[edit]

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 for all ij. The pair (X, πi) must be universal in the sense that for any other such pair (Y, ψi) (i.e. ψi: YXi with ψi = fij o ψj for all ij) there exists a unique morphism u: YX making all the "obvious" identities true; i.e., the diagram

InverseLimit-01.png

must commute for all ij, for which it suffices to show that ψi = πi o u for all i. The inverse limit is often denoted

X = \varprojlim X_i

with the inverse system (Xi, fij) 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 ij if and only if ij. An inverse system is then just a contravariant functor IC. And the inverse limit functor \varprojlim:C^{I^{op}}\rightarrow C is a covariant functor.

Examples[edit]

  • 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.
  • 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^nR[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^nR[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 (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[edit]

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 fij 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 (Ai, fij), (Bi, gij), and (Ci, hij) are three projective systems of abelian groups, and

0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0

is a short exact sequence of inverse systems, then

0\rightarrow\varprojlim A_i\rightarrow\varprojlim B_i\rightarrow\varprojlim C_i\rightarrow\varprojlim{}^1A_i

is an exact sequence in Ab.

Mittag-Leffler condition[edit]

If the ranges of the morphisms of an inverse system of abelian groups (Ai, fij) are stationary, that is, for every k there exists jk such that for all ij : f_{kj}(A_j)=f_{ki}(A_i) one says that the system satisfies the Mittag-Leffler condition. This condition implies that \varprojlim{}^1A_i=0.

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

  • a system in which the morphisms fij 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 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.

Further results[edit]

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^n\varprojlim:C^I\rightarrow 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{}^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 Ai = 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 lim1 Ai ≠ 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 nd + 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[edit]

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[edit]

Notes[edit]

  1. ^ a b c John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.

References[edit]