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:
“These are our grievances which we have thus laid before his majesty with that freedom of language and sentiment which becomes a free people, claiming their rights as derived from the laws of nature, and not as the gift of their chief magistrate.”
—Thomas Jefferson (1743–1826)
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (1894–1963)
“We live in oppressive times. We have, as a nation, become our own thought police; but instead of calling the process by which we limit our expression of dissent and wonder “censorship,” we call it “concern for commercial viability.””
—David Mamet (b. 1947)