A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence Ker(f)⟶Ker(α)⟶Ker(β)⟶Ker(γ)⟶^S coker(α)⟶coker(β)⟶coker(γ)⟶coker(g'). This commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map S is called a connecting homomorphism and describes a curve from the end of the upper row (Ker(γ)⊆C) to the beginning of the lower row (coker(α) = A'/Im(α)), which suggested the name given to this lemma.

commutative diagram | connecting homomorphism | diagram lemma | exact sequence