exact sequence

An exact sequence is a chain complex with no slack: at every spot, everything that maps in is exactly everything that maps to zero further on. Picture water flowing through a pipeline where, at each junction, the water arriving from upstream is precisely the water permitted to continue downstream — nothing pools, nothing leaks. Exactness is the algebraic statement that a sequence of maps fits together perfectly, and it is the language in which most structural theorems are phrased.

Precisely, a sequence of module homomorphisms ... -> A -> B -> C -> ... is exact at B if the image of the map into B equals the kernel of the map out of B. A sequence is exact if it is exact at every term. Since image-contained-in-kernel is the condition d^2 = 0, an exact sequence is exactly a chain complex with all homology zero. Special small cases carry their own names and meaning.

The exactness of 0 -> A -> B says A -> B is injective; the exactness of B -> C -> 0 says B -> C is surjective; exactness of 0 -> A -> B -> 0 says A -> B is an isomorphism. The five-term shape 0 -> A -> B -> C -> 0 is a short exact sequence, which encodes B as built from a sub A and a quotient C. Reading a complicated object through exact sequences — breaking it into known pieces — is one of the central techniques of the subject.