Inverse Limit - Derived Functors of The Inverse Limit

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 f_ij that ensures the exactness of . Specifically, Eilenberg constructed a functor

(pronounced "lim one") such that if (A_i, f_ij), (B_i, g_ij), and (C_i, h_ij) are three projective systems of abelian groups, and

is a short exact sequence of inverse systems, then

is an exact sequence in Ab.