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| In [[mathematics]], the '''inverse limit''' (also called the '''projective limit''') is a construction which allows one to "glue together" several related [[mathematical object|objects]], the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any [[category (mathematics)|category]].
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| == Formal definition ==
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| === Algebraic objects ===
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| We start with the definition of an '''inverse''' (or '''projective''') '''system''' of [[group (mathematics)|groups]] and [[group homomorphism|homomorphisms]]. Let (''I'', ≤) be a [[directed set|directed]] [[poset]] (not all authors require ''I'' to be directed). Let (''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub> be a [[indexed family|family]] of groups and suppose we have a family of homomorphisms ''f''<sub>''ij''</sub>: ''A''<sub>''j''</sub> → ''A''<sub>''i''</sub> for all ''i'' ≤ ''j'' (note the order) with the following properties:
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| # ''f''<sub>''ii''</sub> is the identity on ''A''<sub>''i''</sub>,
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| # ''f''<sub>''ik''</sub> = ''f''<sub>''ij''</sub> <small>o</small> ''f''<sub>''jk''</sub> for all ''i'' ≤ ''j'' ≤ ''k''.
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| Then the pair ((''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub>, (''f''<sub>''ij''</sub>)<sub>''i''≤ ''j''∈''I''</sub>) is called an inverse system of groups and morphisms over ''I'', and the morphisms ''f''<sub>''ij''</sub> are called the transition morphisms of the system.
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| We define the '''inverse limit''' of the inverse system ((''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub>, (''f''<sub>''ij''</sub>)<sub>''i''≤ ''j''∈''I''</sub>) as a particular [[subgroup]] of the [[direct product]] of the ''A''<sub>''i''</sub>'s:
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| :<math>\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\}.</math>
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| The inverse limit, ''A'', comes equipped with ''natural projections'' π<sub>''i''</sub>: ''A'' → ''A''<sub>''i''</sub> which pick out the ''i''th 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.
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| This same construction may be carried out if the ''A''<sub>''i''</sub>'s are [[Set (mathematics)|sets]],<ref name="same-construction">John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.</ref> semigroups,<ref name="same-construction"/> topological spaces,<ref name="same-construction"/> [[ring (mathematics)|rings]], [[module (mathematics)|modules]] (over a fixed ring), [[algebra over a field|algebras]] (over a fixed field), etc., and the [[homomorphism]]s are homomorphisms in the corresponding [[category theory|category]]. The inverse limit will also belong to that category.
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| === General definition ===
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| The inverse limit can be defined abstractly in an arbitrary [[category (mathematics)|category]] by means of a [[universal property]]. Let (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) 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 π<sub>''i''</sub>: ''X'' → ''X''<sub>''i''</sub> (called ''projections'') satisfying π<sub>''i''</sub> = ''f''<sub>''ij''</sub> <small>o</small> π<sub>''j''</sub> for all ''i'' ≤ ''j''. The pair (''X'', π<sub>''i''</sub>) must be universal in the sense that for any other such pair (''Y'', ψ<sub>''i''</sub>) there exists a unique morphism ''u'': ''Y'' → ''X'' making all the "obvious" identities true; i.e., the diagram
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| <div style="text-align: center;">[[Image:InverseLimit-01.png]]</div>
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| must [[commutative diagram|commute]] for all ''i'' ≤ ''j''. The inverse limit is often denoted
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| :<math>X = \varprojlim X_i</math>
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| with the inverse system (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) being understood.
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| 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.
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| 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
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| <math>\varprojlim:C^{I^{op}}\rightarrow C</math> is a [[covariant functor]].
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| == Examples ==
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| * The ring of [[p-adic number|''p''-adic integers]] is the inverse limit of the rings '''Z'''/''p''<sup>''n''</sup>'''Z''' (see [[modular arithmetic]]) with the index set being the [[natural number]]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.
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| * The ring <math>\textstyle R[[t]]</math> of [[formal power series]] over a commutative ring ''R'' can be thought of as the inverse limit of the rings <math>\textstyle R[t]/t^nR[t]</math>, indexed by the natural numbers as usually ordered, with the morphisms from <math>\textstyle R[t]/t^{n+j}R[t]</math> to <math>\textstyle R[t]/t^nR[t]</math> given by the natural projection.
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| * [[Pro-finite group]]s are defined as inverse limits of (discrete) finite groups.
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| * Let the index set ''I'' of an inverse system (''X''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) have a [[greatest element]] ''m''. Then the natural projection π<sub>''m''</sub>: ''X'' → ''X''<sub>''m''</sub> is an isomorphism.
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| * 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'''.
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| ** The set of infinite [[String (computer science)|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 topology|discrete]], the limit space is [[totally disconnected]]. This is one way of realizing the [[p-adic|''p''-adic numbers]] and the [[Cantor set]] (as infinite strings).
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| * Let (''I'', =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the [[product (category theory)|product]].
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| * 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 (category theory)|pullback]].
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| ==Derived functors of the inverse limit==
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| For an [[abelian category]] ''C'', the inverse limit functor
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| :<math>\varprojlim:C^I\rightarrow C</math>
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| is [[Exact functor|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''<sub>''ij''</sub> that ensures the exactness of <math>\varprojlim</math>. Specifically, [[Samuel Eilenberg|Eilenberg]] constructed a functor
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| :<math>\varprojlim{}^1:\operatorname{Ab}^I\rightarrow\operatorname{Ab}</math>
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| (pronounced "lim one") such that if (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>), (''B''<sub>''i''</sub>, ''g''<sub>''ij''</sub>), and (''C''<sub>''i''</sub>, ''h''<sub>''ij''</sub>) are three projective systems of abelian groups, and
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| :<math>0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0</math>
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| is a [[short exact sequence]] of inverse systems, then
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| :<math>0\rightarrow\varprojlim A_i\rightarrow\varprojlim B_i\rightarrow\varprojlim C_i\rightarrow\varprojlim{}^1A_i</math> | |
| is an exact sequence in '''Ab'''.
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| ===Mittag-Leffler condition===
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| If the ranges of the morphisms of the inverse system of abelian groups (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) are ''stationary'', that is, for every ''k'' there exists ''j'' ≥ ''k'' such that for all ''i'' ≥ ''j'' :<math> f_{kj}(A_j)=f_{ki}(A_i)</math> one says that the system satisfies the '''Mittag-Leffler condition'''. This condition implies that <math>\varprojlim{}^1A_i=0.</math>
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| For a discussion of the name "Mittag-Leffler" in its relation with the [[Mittag-Leffler theorem]], see this [http://mathoverflow.net/questions/14717/mittag-leffler-condition-whats-the-origin-of-its-name thread] on [[MathOverflow]].
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| The following situations are examples where the Mittag-Leffler condition is satisfied:
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| * a system in which the morphisms ''f''<sub>''ij''</sub> are surjective
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| * a system of finite-dimensional vector spaces.
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| An example where this is non-zero is obtained by taking ''I'' to be the non-negative [[integer]]s, letting ''A''<sub>''i''</sub> = ''p''<sup>''i''</sup>'''Z''', ''B''<sub>''i''</sub> = '''Z''', and ''C''<sub>''i''</sub> = ''B''<sub>''i''</sub> / ''A''<sub>''i''</sub> = '''Z'''/''p''<sup>''i''</sup>'''Z'''. Then
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| :<math>\varprojlim{}^1A_i=\mathbf{Z}_p/\mathbf{Z}</math>
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| where '''Z'''<sub>''p''</sub> denotes the [[p-adic integers]].
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| ===Further results===
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| More generally, if ''C'' is an arbitrary abelian category that has [[Injective object#Enough injectives|enough injectives]], then so does ''C''<sup>''I''</sup>, and the right [[derived functors]] of the inverse limit functor can thus be defined. The ''n''th right derived functor is denoted
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| :<math>R^n\varprojlim:C^I\rightarrow C.</math>
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| In the case where ''C'' satisfies [[Grothendieck]]'s axiom [[Abelian category#Grothendieck's axioms|(AB4*)]], [[Jan-Erik Roos]] generalized the functor lim<sup>1</sup> on '''Ab'''<sup>''I''</sup> to series of functors lim<sup>n</sup> such that
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| :<math>\varprojlim{}^n\cong R^n\varprojlim.</math>
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| It was thought for almost 40 years that Roos had proved (in ''Sur les foncteurs dérivés de lim. Applications. '') that lim<sup>1</sup> ''A''<sub>''i''</sub> = 0 for (''A''<sub>''i''</sub>, ''f''<sub>''ij''</sub>) 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 lim<sup>1</sup> ''A''<sub>''i''</sub> ≠ 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*)).
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| [[Barry Mitchell (mathematician)|Barry Mitchell]] has shown (in "The cohomological dimension of a directed set") that if ''I'' has [[cardinality]] <math>\aleph_d</math> (the ''d''th [[Aleph number|infinite cardinal]]), then ''R''<sup>''n''</sup>lim 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).
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| == Related concepts and generalizations ==
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| The [[dual (category theory)|categorical dual]] of an inverse limit is a [[direct limit]] (or inductive limit). More general concepts are the [[limit (category theory)|limits and colimits]] of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.
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| ==See also==
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| *[[Direct limit|Direct, or inductive limit]]
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| == Notes ==
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| <references />
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| ==References==
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| *{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Algebra I|publisher=Springer|year=1989|isbn=978-3-540-64243-5|oclc=40551484}}
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| *{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=General topology: Chapters 1-4|publisher=Springer|year=1989|isbn=978-3-540-64241-1|oclc=40551485}}
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| *{{citation|first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=[[Categories for the Working Mathematician]] | edition=2nd |date=September 1998 |publisher=Springer|isbn=0-387-98403-8}}
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| *{{Citation | last=Mitchell | first=Barry | author-link=Barry Mitchell (mathematician) | title=Rings with several objects | journal=[[Advances in Mathematics]] | mr=0294454 | year=1972 | volume=8 | pages=1–161 | doi=10.1016/0001-8708(72)90002-3}}
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| *{{Citation | last=Neeman | first=Amnon | author-link=Amnon Neeman | title=A counterexample to a 1962 "theorem" in homological algebra (with appendix by Pierre Deligne) | journal=[[Inventiones Mathematicae]] | mr=1906154 | year=2002 | volume=148 | issue=2 | pages=397–420 | doi=10.1007/s002220100197}}
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| *{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Sur les foncteurs dérivés de lim. Applications | journal=C. R. Acad. Sci. Paris | mr=0132091 | year=1961 | volume=252 | pages=3702–3704}}
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| *{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Derived functors of inverse limits revisited | journal=[[London Mathematical Society|J. London Math. Soc. (2)]] | mr=2197371 | year=2006 | volume=73 | issue=1 | pages=65–83 | doi=10.1112/S0024610705022416}}
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| * Section 3.5 of {{Weibel IHA}}
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| [[Category:Limits (category theory)]]
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| [[Category:Abstract algebra]]
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| [[de:Limes (Kategorientheorie)]]
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