Orthogonality under a pairing agreeing with θ on continuous classes

Fix a field $k$ and a group $G$ in a common universe, together with a homomorphism $r \colon G \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ (the automorphism group of AlgebraicClosure ℚ over $\mathbb{Q}$), and two $k$-linear representations $M$, $M'$ of $G$. Let pairing be a $k$-bilinear map $H^1(G,M) \times H^1(G,M') \to k$, and let $\theta$ be a $k$-linear map from continuousH1 r M to the $k$-dual of continuousH1 r M'; here continuousH1 r M denotes the submodule of $H^1(G,M)$ obtained as the image under the projection H1π M of the submodule levelCocycles₁ r M of cocycles, and similarly for $M'$. Assume that pairing and $\theta$ agree on these submodules: $\mathrm{pairing}(x)(w) = \theta(x)(w)$ for all $x \in$ continuousH1 r M and $w \in$ continuousH1 r M'. Let $L$ be a $k$-submodule of $H^1(G,M)$ with $L \le$ continuousH1 r M, and let $w \in$ continuousH1 r M'. Then the image of $w$ in $H^1(G,M')$ lies in orthogonal pairing L, that is, in the preimage under the flip of pairing of the dual annihilator of $L$, if and only if $\theta(\langle x, hL\,hx\rangle)(w) = 0$ for every $x \in H^1(G,M)$ and every proof $hx$ that $x \in L$, where $x$ is read as a continuous class via the inclusion $L \le$ continuousH1 r M.

This is the elementary compatibility statement saying that, for local conditions $L$ contained in the continuous part of $H^1$, membership in the orthogonal complement of $L$ under a global pairing depends only on the restricted pairing $\theta$ on continuous classes. It is used in the identification groupCohomology.greenbergWiles_eq_unramifiedMenu_extArithLoc, where orthogonal complements of unramified conditions must be computed through the continuous duality.