Galois cohomology of a finite cyclic group, 1st step

Question

Let $G$ be a finite cyclic group of order $2$ whose generator is $\sigma$. Let $M$ be $G$ module. I want to prove Group cohomology $H^1(G,M)=Z^1(G,M)/B^1(G,M)$ is isomorphic to $ker(\sigma +1)/im(\sigma-1)$.

For the first step, I understand there is a group homomorphism from $Z^1(G,M)$ to $ker(\sigma +1)$. The group homomorphism is $f\to f(\sigma)$ because $(1+\sigma)(f(\sigma))=f(\sigma)+\sigma f(\sigma)=f({\sigma}^2)=f(1)=0$.

But I'm stuck with proving this is bijection.

What is the inverse map of this map?

Thank you for youhr help.

Answer 1

Let $m \in M$ be in the kernel of $\sigma + 1$.

There is a unique cocycle $\xi \in Z^1(G, M)$ with $\xi(1) = 0$ and $\xi(\sigma) = m$ (the only cocycle rule to check is $\xi(\sigma^2) = \sigma \xi(\sigma) + \xi(\sigma)$, but that simplifies to $0 = \sigma m + m$, which is true).

If we modify $m$ by an element of the form $\sigma n - n$, then $\xi$ gets modified by the function $\sigma \mapsto \sigma n - n$, which is a coboundary. So this gives the inverse map $\ker(\sigma+1)/\text{im}(\sigma - 1) \to H^1(G, M)$.