H²_S with cyclotomic twist as tensor invariants

Fix a prime $p$ and a finite set $S$ of rational primes, and let $K, L$ be intermediate fields of $\mathbb{Q}$ in $\overline{\mathbb{Q}} =$ AlgebraicClosure ℚ, with fixing subgroups $\Gamma_K =$ K.fixingSubgroup and $\Gamma_L =$ L.fixingSubgroup inside $\overline{\mathbb{Q}} \simeq_{\mathbb{Q}} \overline{\mathbb{Q}}$. Assume that $\Gamma_L \cap \Gamma_K$, viewed as a subgroup of $\Gamma_K$, is normal and of finite index, and that its relative index in $\Gamma_K$ is coprime to $p$. Let $N$ be a representation of $\Gamma_K$ on a finite-dimensional $\mathbb{Z}/p$-vector space such that $N.\rho(s) = 1$ for every $s \in \Gamma_K$ whose underlying automorphism lies in $\Gamma_L$. The assertion is that the type of $\mathbb{Z}/p$-linear isomorphisms between two spaces is nonempty, i.e. that such an isomorphism exists: on one side, continuousH2Sr for the inclusion $\Gamma_K \hookrightarrow (\overline{\mathbb{Q}} \simeq_{\mathbb{Q}} \overline{\mathbb{Q}})$, the set $S$, and the coefficient module $N$ twisted by the mod $p$ cyclotomic character cycloChar p restricted to $\Gamma_K$ (the twist of a representation $\rho$ by a character $\chi$ sending $g$ to $\chi(g) \cdot \rho(g)$), this being by definition the quotient of levelCocyclesSr₂ by the part of levelCoboundariesSr₂ contained in it; on the other side, the $\Gamma_K$-invariants of the tensor product representation cyclotomicQuotientH2Rep S K L p $\otimes\, N$ in Rep (ZMod p) ↥K.fixingSubgroup.

This is the untwisting step on the $H^2$ side: degree-two $S$-level continuous cohomology of $\Gamma_K$ with coefficients in a cyclotomically twisted finite $\mathbb{Z}/p$-representation that is trivial on $\Gamma_L$ is recovered from the fixed representation cyclotomicQuotientH2Rep S K L p by tensoring with $N$ and taking invariants, the coprimality hypothesis making an averaging argument available. It feeds the subsequent computation of the dimension of this $H^2$ in terms of $p$-torsion in the $S$-class group and local contributions.