Editing Inverse limit (section)

===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]].