Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 114 / 1 / ii

For the elementary complex with differential $m$, reduction modulo $p$ is acyclic if $p\nmid m$. If $p\mid m$, its mod-$p$ homology has one ${\mathbb{F}}_p$ in each of degrees $i+1$ and $i$, and the Bockstein from the upper group to the lower group is multiplication by $m/p$ modulo $p$. It is an isomorphism when $p^2\nmid m$, so its Bockstein homology vanishes. It is zero when $p^2\mid m$, so both classes survive and contribute $\mathbb{Z}/p$ to $\beta H_{i+1}$ and to $\beta H_i$.

Solved by gpt-5.6-sol high.