long exact sequence

A long exact sequence is the bridge that lets you compute the homology of a complicated object from the homologies of simpler pieces. Suppose you have torn a complex into a sub-piece and a quotient-piece. Their homologies do not simply add up, but they are chained together — over and over, in every degree — by a single recurring connecting map, producing one infinite exact sequence that ties all three sets of homology groups together.

The precise statement: a short exact sequence of chain complexes 0 -> A -> B -> C -> 0 gives rise to a long exact sequence ... -> H_n(A) -> H_n(B) -> H_n(C) -> H_{n-1}(A) -> H_{n-1}(B) -> ... in homology, continuing in both directions. The maps H_n(A) -> H_n(B) and H_n(B) -> H_n(C) come from the chain maps; the map H_n(C) -> H_{n-1}(A) is the connecting homomorphism, which drops degree by one. Exactness holds at every term.

The existence of this sequence is the content of the snake lemma applied across the complex, and it is arguably the single most-used computational tool in the subject. In cohomology the sequence runs the other way, raising degree at the connecting map. Whenever an invariant is defined through a complex — homology, cohomology, derived functors like Ext and Tor — a short exact sequence of inputs yields a long exact sequence of outputs, turning unknown groups into known ones.