Notice that in an exact sequence, the composition fi+1 ∘ fi maps Ai to 0 in Ai+2, so every exact sequence is a chain complex. Furthermore, only fi-images of elements of Ai are mapped to 0 by fi+1, so the homology of this chain complex is trivial. More succinctly: Exact sequences are precisely those chain complexes which are acyclic. Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.
Notice that in an exact sequence, the composition fi+1 ∘ fi maps Ai to 0 ,in Ai+2, so every exact sequence is a chain complex. Furthermore only fi-images of elements of Ai are mapped to 0 by fi+1, so the homolog,y of this chain complex is trivial. More succinctly: Exact sequences are precisely those chain complexes which are acyclic. Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.