Level-constancy of the connecting cochain δ¹(c)

Let $k$ be a commutative ring, $G$ a group, and $r \colon G \to \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ a group homomorphism into the $\mathbb{Q}$-algebra automorphisms of AlgebraicClosure ℚ, so that each intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$ gives a level subgroup $r^{-1}(\operatorname{Gal}(\overline{\mathbb{Q}}/F))$. Let $A, B, C$ be $k$-linear representations of $G$ and $\varphi \colon A \to B$, $\psi \colon B \to C$ morphisms of representations, assumed to form a short exact sequence in the pointwise sense: $\varphi$ is injective on underlying modules (hφ), $\psi$ is surjective (hψ), and $\psi(b) = 0$ holds exactly when $b$ lies in the image of $\varphi$ (hex). Assume further that $B$ is smooth pointwise (hsm): every $m \in B$ admits a finite extension $F/\mathbb{Q}$ inside $\overline{\mathbb{Q}}$ with $\rho_B(s)m = m$ for all $s$ such that $r(s)$ fixes $F$ pointwise. Let $c$ be an inhomogeneous $1$-cocycle of $C$ (cocycles₁ C) satisfying IsLevelConstant₁ r c, i.e. for some finite $F/\mathbb{Q}$ one has $c(gs) = c(g)$ whenever $r(s)$ fixes $F$ pointwise. The conclusion is that the connecting $2$-cochain deltaCochain₁ φ ψ hψ c, the $A$-valued cochain obtained by lifting $c$ along the chosen set-theoretic section Function.surjInv hψ of $\psi$, applying the differential d₁₂ of $B$, and taking $\varphi$-preimages, belongs to levelCocycles₂ r A: it is a $2$-cocycle of $A$ and is level-constant in both variables.

This is the cochain-level input for the connecting map $H^1_{\mathrm{cts}}(G,C) \to H^2_{\mathrm{cts}}(G,A)$ attached to a short exact sequence of smooth representations, the well-definedness and exactness statements being separate. It is used in the construction of the linear map on level-constant $1$-cocycles inducing this connecting map, and in the surjectivity statement for the induced map on continuous $H^2$.