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

Apply the Snake lemma degree by degree to a short exact sequence of complexesIf $[c]\in H_n(C)$, lift a cycle $c$ to $b\in B_n$. Its boundary maps to zero in $C_{n-1}$, so it comes from a cycle $a\in A_{n-1}$; define $\partial [c]=[a]$. The Snake-lemma exactness and independence checks yieldThis is the algebraic Mayer-Vietoris theorem for homology objects.