Degree-one Shapiro lemma for continuous H¹

Let $k$ be a commutative ring and $G$ a group (in the same universe), let $r \colon G \to \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ be a group homomorphism into the group of $\mathbb{Q}$-algebra automorphisms of AlgebraicClosure ℚ, and let $S$ be a subgroup of $G$ subject to the hypothesis hS: there is an intermediate field $F_0$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, whose fixing subgroup pulls back along $r$ into $S$ (so $S$ contains a level subgroup, i.e. is open for the topology defined by $r$). Let $N$ be a $k$-linear representation of $S$. For a representation $M$ of a group equipped with a level map $r$, groupCohomology.continuousH1 r M is the submodule of $H^1(M)$ obtained as the image, under the canonical projection H1π from $1$-cocycles to $H^1$, of the submodule levelCocycles₁ r M of level-constant $1$-cocycles. The conclusion asserts that the type of $k$-linear equivalences from continuousH1 r (Rep.coind S.subtype N) to continuousH1 (r.comp S.subtype) N is nonempty; thus the continuous $H^1$ of $G$ on the coinduced representation $\operatorname{CoInd}_S^G N$ and the continuous $H^1$ of $S$ on $N$, the latter taken for the restricted level map $r|_S$, are isomorphic as $k$-modules, although no particular isomorphism is named by the statement.

This is Shapiro's lemma in degree one, in the form appropriate to the continuous (level-constant) cohomology used throughout: coinduction from an open subgroup does not change continuous $H^1$ up to $k$-linear isomorphism. It is used in the local computations of continuous $H^1$, namely in the Euler–Poincaré identity and in the finite-dimensionality of continuous $H^1$ for open subgroups in the prime-local setting.