Restriction to a finite-index subgroup is injective on H¹

Let $k$ be a commutative ring and $G$ a group (in the same universe), let $A$ be a $k$-linear representation of $G$, i.e. an object of Rep k G, and let $S \le G$ be a subgroup of finite index whose index $[G:S]$, viewed in $k$ via the canonical map $\mathbb{N} \to k$, is a unit. Let $c$ be an element of cocycles₁ A, that is, a $1$-cocycle $c : G \to A$ for the $G$-action $\rho$ on $A$ (so $c(gg') = \rho(g)(c(g')) + c(g)$), regarded as a function by coercion. Assume that the restriction of $c$ to $S$ is a coboundary: there exists $a \in A$ with $c(s) = \rho(s)(a) - a$ for every $s \in S$. The conclusion is that $c$ itself is a coboundary on all of $G$: there exists $a \in A$ such that $c(g) = \rho(g)(a) - a$ for every $g \in G$. No normality assumption on $S$ is made, and the statement is at the level of cochains rather than cohomology classes.

This is the injectivity of the restriction map $H^1(G,A) \to H^1(S,A)$ when the index $[G:S]$ is invertible in the coefficient ring, stated on representatives. It is used to compare $H^1$ of a decomposition group with that of a subgroup in the local bridge lemmas NumberField.PlaceDecomp.exists_unit_inv_map_delta_res_eq_theta_localBridge and its primary variant, and in groupCohomology.exists_linearEquiv_H1_of_forall_iff_of_isUnit_index.