A diagram lemma which states that every short exact sequence of chain complexes and chain homomorphisms 0⟶C⟶^ϕ D⟶^ψ ℰ⟶0 gives rise to a long exact sequence in homology ...⟶H_p(C)⟶^(Φ_*) H_p(D)⟶^(Ψ_*) H_p(ℰ)⟶^(d_*) H_(p - 1)(C)⟶^(Φ_*) H_(p - 1)(D)⟶..., where the map d_* is the chain homomorphism induced by the boundary operator of the chain complex D. The name of this lemma is due to its proof, which consists of diagram chasing along a staircase-like path.

chain complex | chain homomorphism | diagram chasing | diagram lemma | exact sequence | homology