Exactness at C^G of the connecting sequence

Let $k$ be a commutative ring and $G$ a group, and let $A$, $B$, $C$ be representations of $G$ over $k$, with morphisms of representations $\varphi : A \to B$ and $\psi : B \to C$. Assume the underlying $k$-linear map of $\varphi$ is injective, the underlying map of $\psi$ is surjective, and for every $b \in B$ one has $\psi(b) = 0$ if and only if $b = \varphi(a)$ for some $a \in A$. Let $c \in C$ be invariant, i.e. $\rho_C(g)c = c$ for all $g \in G$. Write $\sigma$ for the set-theoretic section of $\psi$ determined by the surjectivity hypothesis, and let $\delta^0(c)$ be the connecting $1$-cochain $G \to A$ characterised by $\varphi(\delta^0(c)(g)) = \rho_B(g)(\sigma c) - \sigma c$. The assertion is an equivalence: $\delta^0(c)$ lies in the submodule of $1$-coboundaries of $A$, that is, there is $a_0 \in A$ with $\delta^0(c)(g) = \rho_A(g)a_0 - a_0$ for all $g$, if and only if there exists an invariant $b \in B$ with $\psi(b) = c$.

This is exactness of $B^G \to C^G \to H^1(G,A)$ at $C^G$ for a short exact sequence of $G$-representations, in the explicit cochain form needed later: the class of $\delta^0(c)$ vanishes exactly when $c$ lifts to an invariant of $B$. No continuity, smoothness or level structure enters at this point; the result is used in the construction of the connecting map on continuous cohomology and its exactness properties, being cited by groupCohomology.bijective_theta_of_shortExact, groupCohomology.continuousH2Map_kummerRep_injective_and_range_iff_smul_eq_zero and groupCohomology.finrank_euler_even_eq_odd_of_continuousH2MapHom_surjective.