Editing Inverse limit (section)

==Derived functors of the inverse limit==

For an [[abelian category]] ''C'', the inverse limit functor
:<math>\varprojlim:C^I\rightarrow C</math>
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
:<math>\varprojlim{}^1:\operatorname{Ab}^I\rightarrow\operatorname{Ab}</math>
(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 inverse systems of abelian groups, and
:<math>0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0</math>
is a [[short exact sequence]] of inverse systems, then
:<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'''.

===Mittag-Leffler condition===

If the ranges of the morphisms of an 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'''.

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''<sub>''ij''</sub> are surjective
* a system of finite-dimensional [[vector space]]s or finite abelian groups or modules of finite [[length of a module|length]] or [[Artinian module]]s.

An example where <math>\varprojlim{}^1</math> 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
:<math>\varprojlim{}^1A_i=\mathbf{Z}_p/\mathbf{Z}</math>
where '''Z'''<sub>''p''</sub> denotes the [[p-adic integers]].

===Further results===

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 functor]]s of the inverse limit functor can thus be defined. The ''n''th right derived functor is denoted
:<math>R^n\varprojlim:C^I\rightarrow C.</math>
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
:<math>\varprojlim{}^n\cong R^n\varprojlim.</math>
It was thought for almost 40 years that Roos had proved (in {{lang|fr|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*)).

[[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<sup>''n''</sup>, on diagrams indexed by a countable set, is nonzero for&nbsp;''n''&nbsp;>&nbsp;1).