An infinite sequence of homomorphisms of modules or additive Abelian groups ...->C_(i + 1) ->^(d_(i + 1)) C_i ->^(d_i) C_(i - 1)->... such that, for all indices i element Z, Im(d_(i + 1)) = Ker(d_i). In other words, a long exact sequence is a chain complex such that for all i, the ith homology module (or group) H^i(C) = Ker(d_i)/Im(d_(i + 1)) is the zero module (or the zero group).

chain complex | homology | long exact sequence of a pair axiom | short exact sequence