Inverse limit
Encyclopedia
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
.
and homomorphisms
. Let (I, ≤) 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 fij: Aj → Ai for all i ≤ j (note the order) with the following properties:
Then the pair ((Ai)i∈I, (fij)i≤ j∈I) 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)i∈I, (fij)i≤ j∈I) as a particular subgroup
of the direct product
of the Ai's:
The inverse limit, A, comes equipped with natural projections πi: A → Ai 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, rings
, modules
(over a fixed ring), algebras
(over a fixed field), etc., and the homomorphism
s are homomorphisms in the corresponding category
. The inverse limit will also belong to that category.
by means of a universal property
. Let (Xi, fij) be an inverse system of objects and morphism
s 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 = fij o 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 making all the "obvious" identities true; i.e., the diagram
must commute
for all i ≤ j. The inverse limit is often denoted
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 functor
s. 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
is a covariant functor.
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
such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three projective systems of abelian groups, and
is a short exact sequence of inverse systems, then
is an exact sequence in Ab.
one says that the system satisfies the Ḿittag-Leffler condition. This condition implies that 
For a discussion of the name "Mittag-Leffler" in its relation with the Mittag-Leffler theorem, see this thread on MathOverflow.
The following situations are examples where the Mittag-Leffler condition is satisfied :
An example where this is non-zero is obtained by taking I to be the non-negative integer
s, letting Ai = piZ, Bi = Z, and Ci = Bi / Ai = Z/piZ. Then
where Zp denotes the p-adic integers.
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 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
(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 lim^n, on diagrams indexed by a countable set, is nonzero for n>1).
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
.
Algebraic objects
We start with the definition of an inverse (or projective) system of groupsGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and homomorphisms
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
. Let (I, ≤) be a directed
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
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 fij: Aj → Ai for all i ≤ j (note the order) with the following properties:
- fii is the identity in Ai,
- fik = fij o fjk for all i ≤ j ≤ k.
Then the pair ((Ai)i∈I, (fij)i≤ j∈I) 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)i∈I, (fij)i≤ j∈I) as a particular subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of the direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of the Ai's:
The inverse limit, A, comes equipped with natural projections πi: A → Ai 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
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
described in the next section.
This same construction may be carried out if the Ai's are sets, rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
(over a fixed ring), algebras
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
(over a fixed field), etc., and the homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s are homomorphisms in the corresponding category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. The inverse limit will also belong to that category.
General definition
The inverse limit can be defined abstractly in an arbitrary categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
by means of a universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
. Let (Xi, fij) be an inverse system of objects and morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s 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 = fij o 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 making all the "obvious" identities true; i.e., the diagram
must commute
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
for all i ≤ j. The inverse limit is often denoted
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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
X′ → X commuting with the projection maps.
We note that an inverse system in a category C admits an alternative description in terms of functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
i ≤ j. An inverse system is then just a contravariant functor I → C. And the inverse limit functor
is a covariant functor.Examples
- The ring of p-adic integersP-adic numberIn mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
is the inverse limit of the rings Z/pnZ (see modular arithmeticModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
) with the index set being the natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s 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
of formal power seriesFormal power seriesIn mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
over a commutative ring R can be thought of as the inverse limit of the rings
, indexed by the natural numbers as usually ordered, with the morphisms from 
to 
given by the natural projection. - Pro-finite groupPro-finite groupIn mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
s are defined as inverse limits of (discrete) finite groups. - Let the index set I of an inverse system (Xi, fij) have a greatest elementGreatest elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
m. Then the natural projection πm: X → Xm is an isomorphism. - Inverse limits in the category of topological spacesCategory of topological spacesIn mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
are given by placing the initial topologyInitial topologyIn general topology and related areas of mathematics, the initial topology on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The subspace topology and product topology constructions are both special cases of initial...
on the underlying set-theoretic inverse limit. This is known as the limit topology.- The set of infinite stringsString (computer science)In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
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 setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
(as infinite strings).
- The set of infinite strings
- Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the productProduct (category theory)In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
. - 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 pullbackPullback (category theory)In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
.
Derived functors of the inverse limit
For an abelian categoryAbelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
C, the inverse limit functor
is left exact
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
. 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, EilenbergSamuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
constructed a functor
such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three projective systems of abelian groups, and
is a short exact sequence of inverse systems, then
is an exact sequence in Ab.
Mittag-Leffler condition
If the ranges of the morphisms of the 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 Ḿittag-Leffler condition. This condition implies that For a discussion of the name "Mittag-Leffler" in its relation with the Mittag-Leffler theorem, see this thread on MathOverflow.
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 integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, letting Ai = piZ, Bi = Z, and Ci = Bi / Ai = Z/piZ. Then
where Zp denotes the p-adic integers.
Further results
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 denotedIn 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 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
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
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
(the dth infinite cardinalAleph number
In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph...
), 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 lim^n, on diagrams indexed by a countable set, is nonzero for n>1).
Related concepts and generalizations
The categorical dualDual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
of an inverse limit is a direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
(or inductive limit). More general concepts are the limits and colimits
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.