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 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.

Read more about this topic:  Inverse Limit

Famous quotes containing the words derived, inverse and/or limit:

    “The sceptics assert, though absurdly, that the origin of all religious worship was derived from the utility of inanimate objects, as the sun and moon, to the support and well-being of mankind.”
    —David Hume (1711–1776)

    “The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
    —Aldous Huxley (1894–1963)

    “Greatness collapses of itself: such limit the gods have set to the growth of prosperous states.”
    —Marcus Annaeus Lucan (39–65)