A <complex in an abelian category> is a sequence $(A_n,d_n)$ with $d_nd_{n+1}=0$. A sequence is <exact sequence in an abelian category>[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.

The <Snake lemma> associates to a morphism of short exact sequences the exact sequence
$$
\ker f'\to\ker f\to\ker f''
\xrightarrow{\partial}
\operatorname{coker}f'\to\operatorname{coker}f\to\operatorname{coker}f''.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 6 / a / Solution

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.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 6 / c / Solution

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In a commutative diagram of two exact five-term sequences, if the first and fourth vertical maps are epimorphisms and the second and fifth are monomorphisms, then an isomorphism in the middle follows; in particular, four surrounding isomorphisms force the fifth.


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