Exactness at H¹ for level-constant continuous cochains

Let $k$ be a commutative ring, $G$ a group and $r\colon G\to\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$ a group homomorphism into the automorphism group of AlgebraicClosure ℚ over $\mathbb Q$, and let $\varphi\colon A\to B$, $\psi\colon B\to C$ be morphisms of $k$-linear representations of $G$ such that the underlying map of $\varphi$ is injective, that of $\psi$ is surjective, and $\psi(b)=0$ holds exactly when $b=\varphi(a)$ for some $a\in A$. Assume further that $B$ is smooth pointwise: each $m\in B$ admits a finite subextension $F$ of $\overline{\mathbb Q}/\mathbb Q$ with $B.\rho(s)m=m$ for every $s$ with $r(s)$ in the fixing subgroup of $F$. Let $c$ be an inhomogeneous $1$-cocycle of $C$ satisfying the predicate IsLevelConstant₁ r (existence of a finite subextension $F/\mathbb Q$ such that the cochain is unchanged when its argument is multiplied by an element of $r^{-1}$ of the fixing subgroup of $F$). Then the connecting $2$-cochain deltaCochain₁ φ ψ hψ c, the $A$-valued cochain whose image under $\varphi$ is $d^1$ of a set-theoretic $\psi$-lift of $c$, lies in levelCoboundaries₂ r A, i.e. equals $d^1e$ for some level-constant $1$-cochain $e\colon G\to A$, if and only if there is a level-constant $1$-cocycle $b$ of $B$ with $c-\psi\circ b$ an ordinary $1$-coboundary of $C$.

This is exactness of the long exact sequence of continuous (level-constant) cohomology at $H^1_{\mathrm{cts}}(G,C)$, in the cochain-level form: the class of the connecting cochain $\delta^1(c)$ vanishes in $H^2_{\mathrm{cts}}(G,A)$ precisely when $c$ comes from a class in $H^1_{\mathrm{cts}}(G,B)$. It is used in the analysis of the maps on continuous $H^2$ attached to a short exact sequence of representations, in particular for groupCohomology.bijective_theta_of_shortExact and for the injectivity and image computations for the Kummer representation.