(A_{n}, d_{n})

with

d_{n} d_{n + 1} = 0

. A sequence is exact when the image of every incoming map equals the kernel of the outgoing map.

The Five lemma says that in a morphism between exact five-term sequences, suitable epimorphism assumptions on the left and monomorphism assumptions on the right, together with isomorphisms in the four surrounding positions, force the middle map to be an isomorphism.

ker f^{'} \to ker f \to ker f^{''} \partial coker f^{'} \to coker f \to coker f^{''} .

b

Solution

Apply the Snake lemma degree by degree to a short exact sequence of complexes

0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0.

[c] \in H_{n} (C)

, lift a cycle

c

b \in B_{n}

. Its boundary maps to zero in

C_{n - 1}

, so it comes from a cycle

a \in A_{n - 1}

\partial [c] = [a]

. The Snake-lemma exactness and independence checks yield

\dots \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \partial H_{n - 1} (A) \to H_{n - 1} (B) \to \dots .

The commutative diagram of short exact sequences of complexes induces a commutative diagram between the two long exact homology sequences from part (b). If any two vertical chain maps induce isomorphisms in every degree, then in each five-term window four of the five vertical homology maps are isomorphisms. The Five lemma makes the remaining map an isomorphism. Rotating the window handles each of the three possible missing columns, proving the two-out-of-three assertion.

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