Two-sided nondegeneracy of a bijective pairing into the dual

Let $k$ be a field and $G$ a group, both in a fixed universe, and let $r : G \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be a group homomorphism into the automorphism group of AlgebraicClosure ℚ over $\mathbb{Q}$. Let $M$ and $D$ be $k$-linear representations of $G$, and write $\mathrm{H}^1_{\mathrm{cont}}(r,M)$ for continuousH1 r M, the $k$-submodule of the first group cohomology $\mathrm{H}^1(G,M)$ obtained as the image, under the canonical map H1π from one-cocycles to $\mathrm{H}^1(G,M)$, of the submodule levelCocycles₁ r M of cocycles attached to $r$; likewise for $D$. Let $\theta : \mathrm{H}^1_{\mathrm{cont}}(r,M) \to \mathrm{Hom}_k(\mathrm{H}^1_{\mathrm{cont}}(r,D),k)$ be a $k$-linear map into the $k$-dual of $\mathrm{H}^1_{\mathrm{cont}}(r,D)$, and assume $\theta$ is bijective. The conclusion is the conjunction of two nondegeneracy statements for the associated pairing $(x,w) \mapsto \theta(x)(w)$: first, any $x \in \mathrm{H}^1_{\mathrm{cont}}(r,M)$ with $\theta(x)(w) = 0$ for all $w \in \mathrm{H}^1_{\mathrm{cont}}(r,D)$ vanishes; second, any $w \in \mathrm{H}^1_{\mathrm{cont}}(r,D)$ with $\theta(x)(w) = 0$ for all $x \in \mathrm{H}^1_{\mathrm{cont}}(r,M)$ vanishes. Nothing beyond the $k$-module structures is used: $r$, $M$ and $D$ serve only to name the two spaces.

This is the standard passage from a bijective map into a dual space to nondegeneracy of the corresponding bilinear pairing, specialised to the pair of continuous first cohomology groups used in the duality bookkeeping of Greenberg–Wiles type Euler-characteristic formulas; the field hypothesis is what makes the second half hold. It is cited in the identification of the Greenberg–Wiles expression with the unramified local menu, groupCohomology.greenbergWiles_eq_unramifiedMenu_extArithLoc.