Connecting 0-cochain is a level-constant 1-cocycle

Let $k$ be a commutative ring, $G$ a group, and $r \colon G \to \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}}) = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ a group homomorphism, where $\overline{\mathbb{Q}}$ is AlgebraicClosure ℚ. Let $A, B, C$ be $k$-linear representations of $G$ and let $\varphi \colon A \to B$, $\psi \colon B \to C$ be morphisms of representations such that $\varphi$ is injective on underlying modules, $\psi$ is surjective, and $\psi(b) = 0$ holds exactly when $b = \varphi(a)$ for some $a \in A$; assume further that $B$ is pointwise smooth for $r$, i.e. for every $m \in B$ there is an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, with $\rho_B(s)m = m$ for all $s \in G$ whose image $r(s)$ lies in the fixing subgroup of $F$. Let $c \in C$ be $G$-invariant. Then the connecting $0$-cochain $\delta^0(c) \colon G \to A$, characterised by $\varphi(\delta^0(c)(g)) = \rho_B(g)\sigma c - \sigma c$ for the chosen set-theoretic section $\sigma$ of $\psi$, is a $1$-cocycle, i.e. $\delta^0(c)(gh) = \rho_A(g)\delta^0(c)(h) + \delta^0(c)(g)$, and it is level-constant in the sense of IsLevelConstant₁: there is an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, such that $\delta^0(c)(gs) = \delta^0(c)(g)$ for all $g \in G$ and all $s \in G$ with $r(s)$ in the fixing subgroup of $F$.

This is the construction of the connecting map $C^G \to H^1(G,A)$ for a short exact sequence of $G$-representations, in the form needed for continuous (level-constant) cohomology of a group mapping to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$: the naive connecting cochain of an invariant class already lands in the level-constant part. It is used in the construction and analysis of the continuous long exact sequence, in particular by the results on bijectivity of the comparison map $\theta$ for short exact sequences, on finite-dimensionality of continuous cohomology, and on the Kummer representation in degree two.