Injectivity of Inflation Homomorphism $H^1(G/K,A^K)\rightarrow H^1(G,A)$ 
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 ?
 A: See page 23 of these notes. It shows a method to show injectivity of inflation.
A: 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 $\pi(\sigma) \in (G/K)^n$.

