The p-torsion of H²_{cts}(Gal(ℚ̄_q/K),ℚ̄_q^×) has order p

Fix a prime $q$ and an intermediate field $K$ of the extension $\mathbb{Q}_q \subseteq \overline{\mathbb{Q}}_q$ (the algebraic closure being PadicAlgCl q) with $K/\mathbb{Q}_q$ finite, and let $r$ be a group homomorphism from $\mathrm{Gal}(\overline{\mathbb{Q}}_q/K)$, realised as the $K$-algebra automorphisms of $\overline{\mathbb{Q}}_q$, to the $\mathbb{Q}$-algebra automorphisms of $\overline{\mathbb{Q}} =$ AlgebraicClosure ℚ. Two compatibility hypotheses are imposed on $r$: for every intermediate field $E$ of $\overline{\mathbb{Q}}_q/K$ with $E/K$ finite there is an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$ with $F/\mathbb{Q}$ finite such that every $\sigma$ with $r\sigma$ in the fixing subgroup of $F$ lies in the fixing subgroup of $E$; and conversely, for every such $F$ there is such an $E$ with $\sigma$ in the fixing subgroup of $E$ implying $r\sigma$ in the fixing subgroup of $F$. Let $p$ be a further prime. Consider the group continuousH2 r (Rep.ofAlgebraAutOnUnits K (PadicAlgCl q)), i.e. the quotient of the submodule levelCocycles₂ of $2$-cocycles satisfying the level condition attached to $r$ by its intersection with the submodule levelCoboundaries₂ of $2$-coboundaries, for the module of units of $\overline{\mathbb{Q}}_q$ with its Galois action. The assertion is that its $p$-torsion submodule, as a $\mathbb{Z}$-module, has exactly $p$ elements.

This is the statement that the $p$-torsion of the Brauer group $\mathrm{Br}(K) = H^2_{\mathrm{cts}}(\mathrm{Gal}(\overline{\mathbb{Q}}_q/K), \overline{\mathbb{Q}}_q^\times)$ of a finite extension $K$ of $\mathbb{Q}_q$ is cyclic of order $p$, for the model of the continuous $H^2$ built from cocycles satisfying a level condition transported through $r$. It feeds the computation that this $H^2$ has $\mathbb{Z}/p$-rank one and the local-global comparison of $p$-torsion classes used in the construction of the invariant map.