(Very abstract algebra, category theory):
An exact sequence is a sequence of objects (which need to come from a special kind of category, a cartesian category; just ignore that and take them to be groups or rings or modules or something else you can imagine) and homomorphisms between them:

1=A0 → A1 → A2 → ... → An → 1
- OR -
1=A0 → A1 → A2 → ... → An → ...
(That is, if it's finite the last element must be 1).

By abuse of notation, we write "1" for the trivial object (the group consisting of just the identity element, the ring {0,1}, etc.).

For the sequence to be exact, we additionally require of the homomorphisms that Im(An-1->An) = Ker(An->An+1).

This has immediate repercussions; see short exact sequence for the most important example.

Homology and cohomology "operators" (really sequences of functors!) transform short exact sequences into infinite exact sequences; this is commonly known as the snake lemma.

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