Isomorphic representations give equivalent S-restricted H¹, H²

Fix a prime $p$, a finite set $S$ of rational primes, and an intermediate field $K$ of $\mathbb{Q}$ inside $\mathrm{AlgebraicClosure}\ \mathbb{Q}$, and write $r$ for the inclusion $K.\mathrm{fixingSubgroup} \hookrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, i.e. the monoid homomorphism K.fixingSubgroup.subtype. Let $A$ and $B$ be representations of the group $K.\mathrm{fixingSubgroup}$ over $\mathbb{Z}/p$, in the sense of objects of Rep (ZMod p) ↥K.fixingSubgroup in Type 0, and let $e : A \cong B$ be an isomorphism of such representations. The conclusion is the conjunction of two nonemptiness assertions. First, the $\mathbb{Z}/p$-module continuousH1Sr r S A, namely the submodule of $H^1(K.\mathrm{fixingSubgroup}, A)$ obtained as the image under the projection H1π A of the submodule levelCocyclesSr₁ r S A of $1$-cocycles, is linearly equivalent over $\mathbb{Z}/p$ to the corresponding submodule for $B$. Second, the $\mathbb{Z}/p$-module continuousH2Sr r S A, namely the quotient of levelCocyclesSr₂ r S A by the preimage of levelCoboundariesSr₂ r S A under the inclusion of that cocycle submodule, is linearly equivalent over $\mathbb{Z}/p$ to the corresponding quotient for $B$. Both equivalences are asserted only as nonemptiness of the type of linear equivalences, with no compatibility with $e$ recorded.

This is the functoriality of the $S$-restricted cohomology modules over the base field $K$ in the coefficient representation: isomorphic coefficients give isomorphic $H^1$ and $H^2$, in the restricted form used in the Euler-characteristic bookkeeping. It is used in establishing finite-dimensionality and the dimension formula for the restricted $H^2$ of a coinduced module, groupCohomology.finiteDimensional_continuousH2S_coind_and_finrank_eq, where coefficients may be replaced by an isomorphic copy.