An exact sequence is a tool used in Algebraic Topology used to extract information from a sequence of chain groups.
Given a sequence of groups and homomorphisms
is an exact sequence if for all , the sequence can be infinite.
Given an exact sequence of chain groups, with this indexing
we have a chain complex.
Short Exact Sequence
Given the special case where we have 3 groups with the following homomorphisms
where is a one-one homomorphism and is an onto homomorphism, we have a short exact sequence. Short exact sequences have the property .