Inflated classes are those with a cocycle vanishing on N

Let $k$ be a commutative ring, $G$ a group, $M$ an object of Rep k G, that is a $k$-linear representation of $G$, and $N$ a normal subgroup of $G$; no triviality assumption is made on the action of $N$ on $M$. Let $x$ be an element of $H^1(G,M)$, in the form H1 M. The theorem asserts an equivalence. On one side, $x$ belongs to the $k$-submodule inflationImage M N of H1 M, defined as the range of the $k$-linear map underlying the inflation morphism $H^1(G/N, M^N) \to H^1(G,M)$, the latter being groupCohomology.map in degree $1$ applied to the quotient homomorphism $G \to G/N$ together with the morphism of representations obtained from the lift of $\rho$ to an action of $G/N$ on the $N$-invariants $M^N$. On the other side, there exists a $1$-cocycle $c$ of $G$ with values in $M$, an element of cocycles₁ M, whose cohomology class H1π M c is equal to $x$ and which satisfies $c(n) = 0$ for every $n \in N$.

This is the cocycle-level description of the image of inflation: a class comes from $H^1(G/N, M^N)$ exactly when it admits a representative cocycle vanishing identically on $N$, which is the usual consequence of the inflation–restriction exact sequence. It is used in the comparison of inflation images for different subgroups and in converting continuity and unramifiedness conditions on classes of a Galois group into membership in inflation images from finite levels, as in groupCohomology.exists_cocycles1_unramified_iff_mem_inflationImage_sup, groupCohomology.inflationImage_eq_inflationImage_of_forall_pow_mem and groupCohomology.invariants_add_dualTwist_le_finrank_continuousClasses.