An exact sequence is a sequence of maps α_i :A_i->A_(i + 1) between a sequence of spaces A_i, which satisfies Im(α_i) = Ker(α_(i + 1)), where Im denotes the image and Ker the group kernel. That is, for a element A_i, α_i(a) = 0 iff a = α_(i - 1)(b) for some b element A_(i - 1). It follows that α_(i + 1) °α_i = 0. The notion of exact sequence makes sense when the spaces are groups, modules, chain complexes, or sheaves. The notation for the maps may be suppressed and the sequence written on a single line as ...->A_(i - 1)->A_i->A_(i + 1)->....

chain complex | homology | long exact sequence | short exact sequence | spectral sequence