Deriving Long Exact Sequences from Short Ones

3 Deriving Long Exact Sequences from Short Ones 

As we have already noted, passing to homology is bound to make things more complex, since, for example, injective maps may cease to be injective. It turns out that instead of one short exact sequence, the relationship between Hn A), Hn B), and Hn B/A) is best described as a long exact sequence, which involves the groups H A), H, B), and H B/A) in all dimensions. 

By our discussion above, we may assume that. 4 is a subcomplex of B, and replace C with B/A. We do that to be able to phrase the proof in simple words. Furthermore, we have already checked that all the connecting homomorphisms are well-defined, so it remains only to check the exactness of the sequence (5.5). 

We also note that by Proposition 5.3, any map between short exact sequences 

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