Derived Functors of The Inverse Limit
For an abelian category 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
(pronounced "lim one") 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.
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