Injectivity of Inflation Homomorphism $H^1(G/K,A^K)\rightarrow H^1(G,A)$

Question

I want to prove that injectivity of $inflation\ homomorphism$ $$0\rightarrow H^1(G/K,A^K)\rightarrow H^1(G,A)$$ where $K$ is normal.

Proof : (a) If $x\in A^K$ then $gH\cdot x:= gx$ This is well-defined : For any $g\in G,\ h\in K$, we have $$ghx=gx,\ hgx=gh'x=gx$$ So $A^K$ is $G/K$-module.

(b) If $\psi : A^K\rightarrow A$ is inclusion and $\phi : G\rightarrow G/K$ is projection, then they are $compactible$ so that the homomorphism is well-defined.

But to prove injectivity, I have no idea. How can we finish the proof ?

Answer 1

See page 23 of these notes. It shows a method to show injectivity of inflation.

Answer 2

Denote $\sigma = (\sigma_1,...,\sigma_n) \in G^n$. Let $\pi : G \to G/K$ and $i : A^K \to A$. Let $\bar{a},\bar{b} \in Z^n(G,A)$ such that $$ \bar{a}_{\sigma} = i(a)_{\pi(\sigma)} \ , \quad \bar{b}_{\sigma} = i(b)_{\pi(\sigma)} $$ Clearly, $\bar{a},\bar{b} \in i(A^K)$. If $\bar{a}=\bar{b}$ then $$ \bar{a}_\sigma - \bar{b}_\sigma = (\partial \bar{c})_\sigma \in A^K $$ I claim that $$ (\overline{a-b})_\sigma = (\partial \bar{c})_\sigma = (\overline{\partial c})_\sigma $$ for $c$ that satisfies $\bar{c}_{\tau} = i(c)_{\pi(\tau)}$. Then we have, as $i$ is injective, $$ a_{\pi(\sigma)} - b_{\pi(\sigma)} = (\partial c)_{\pi(\sigma)} $$ and thus $a - b = \partial c$ which proves $[a]=[b]$ in $Z^1(G/K, A^K)$.

Details to check:

• That $a,b$ are cocycles in $Z^n(G/K, A^K)$.
• That $\partial(\bar{c}) = \overline{ \partial c}$
• That $\bar{x}_\sigma = i(x)_{\pi(\sigma)}$ is well-defined map and does not depend on the representatives $\sigma \in G^n$ of $\pi(\sigma) \in (G/K)^n$.