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A sequence of abelian groups and homomorphisms where the image of one homomorphism is exactly the kernel of the next.

A short exact sequence consists of five terms, starting and ending with the trivial group, illustrating a relationship between a subgroup and a quotient group.

A sequence of abelian groups and homomorphisms that is exact and potentially infinite in length on either or both sides.

Homology describes the algebraic study of the structure of topological spaces through homology groups, which are derived from the chains, cycles, and boundaries within the space.

A chain complex is a sequence of abelian groups connected by homomorphisms that compose to give zero, used to define homology groups.

Cohomology is a way of assigning algebraic invariants to topological spaces, similar to homology but with cochains, cocycles, and coboundaries.

In the context of a homomorphism, the kernel is the set of elements that map to the zero element in the codomain, reflecting the 'roots' of the homomorphism.

The image of a homomorphism is the set of all elements that can be obtained as outputs from the homomorphism, reflecting the 'range' of the homomorphism.

A result in homological algebra stating that, under certain conditions, if four of five terms in two related short exact sequences are isomorphisms, then so is the fifth term.