Local triviality at Q gives classes unramified outside S

Let $k$ be a commutative ring, let $S$ and $Q$ be finite sets of rational primes, and let $M$ be a $k$-linear representation of the group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of ring automorphisms of AlgebraicClosure ℚ over $\mathbb{Q}$. Assume $M$ is smooth, in the sense that every $m \in M$ admits an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$, finite-dimensional over $\mathbb{Q}$, with $\rho(s)m = m$ for all $s$ in the fixing subgroup of $F$; and assume $M$ is unramified outside $S$, in the sense that for every prime $q \notin S$ and every valuation subring $A$ of $\overline{\mathbb{Q}}$ with $q$ a non-unit of $A$, every element of the inertia subgroup of $A$ over $\mathbb{Q}$ (the image in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of the inertia subgroup inside the decomposition subgroup) acts on $M$ as the identity. Let $x \in H^1(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), M)$ lie in continuousH1S (S ∪ Q) M, the image under the projection H1π M of the submodule levelCocyclesS₁ (S ∪ Q) M of $1$-cocycles. Suppose that for every $q \in Q$ the degree-one map on cohomology induced by the homomorphism primeLocalToGlobal q from $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ (restriction of scalars to $\mathbb{Q}$ followed by restriction of normal automorphisms to $\overline{\mathbb{Q}}$), taken with the identity morphism of the restricted representation, annihilates $x$. Then $x \in$ continuousH1S S M.

This is the bookkeeping step which shows that a class of level $S \cup Q$ whose restriction to the local Galois group at each auxiliary prime $q \in Q$ vanishes is already of level $S$; combined with the characterisation of continuousH1S S M as the classes of locally constant cocycles that are coboundaries on all inertia subgroups above primes outside $S$, it identifies the Selmer conditions at the augmented level used in the Taylor–Wiles argument. It is used in the construction of sets of Taylor–Wiles primes, in ResidualGaloisRep.exists_taylorWilesPrimes_card_eq_finrank_continuousH1S_dualTwist.