The class of m· x is m times the class of x

Let $G$ be a group and let $A$ be a representation of $G$ over $\mathbb{Z}$, i.e. an object of Rep ℤ G; let $n$ be a natural number and $m$ an integer. Let $x \colon (\mathrm{Fin}\ n \to G) \to A$ be an inhomogeneous $n$-cochain, and suppose that the degree-$n$ differential of the inhomogeneous cochain complex of $A$ annihilates $x$, so that $x$ is an $n$-cocycle, and that it likewise annihilates $m \cdot x$ (this second hypothesis, although a formal consequence of the first by additivity of the differential, is taken as a separate argument so that the cocycle $m \cdot x$ can be formed). Write cocyclesMk for the passage from a cochain together with a proof that the differential kills it to the corresponding element of the module of $n$-cocycles, and $\pi$ for the quotient map from $n$-cocycles to $H^n(G, A)$. The assertion is that the class of $m \cdot x$ in $H^n(G, A)$ equals $m$ times the class of $x$, i.e. $\pi_{A,n}(\mathrm{cocyclesMk}(m \cdot x)) = m \cdot \pi_{A,n}(\mathrm{cocyclesMk}(x))$.

This records the $\mathbb{Z}$-linearity of the class map on cocycles, in the concrete form in which cocycles are presented by raw inhomogeneous cochains together with a vanishing hypothesis. It is used in the idelic torsion step NumberField.SIdele.exists_smul_eq_d_add_diag_of_d_eq_diag, where one must multiply an explicit cocycle by an integer and track the effect on its cohomology class.