Editing Inverse limit (section)

===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).