Proposition 6

The mapping $c_{\sigma}$ is an element of the group $Z^2(G,F)$, and its class in the cohomology group $H^2(G,F)$ depends only on the class of the $\tau$-extension $\mathscr{E}$.

We say that the mapping $c_{\sigma}$ is the 2-cocycle associated with $\sigma$ and that its cohomology class in $H^2(G,F)$ is the cohomology class of the $\tau$-extension $\mathscr{E}$.

Let us first verify that $c_{\sigma}$ satisfies the cocycle condition. Let $g_1,g_2$, and $g_3$ be elements of G. Using formula (1) of VIII, p. 285 and (9), we obtain the relations

and

The mapping $c_{\sigma}$ is therefore an element of $Z^2(G,F)$.

Let us now prove that the image of $c_{\sigma}$ in $H^2(G,F)$ is independent of the chosen section $\sigma$. Let $\sigma$ and $\sigma '$ be such sections. For every element $g$ of G, there exists a unique element $a_g$ of F such that $\sigma '(g) =\iota (a_g)\sigma (g)$. Let $g_1$ and $g_2$ be elements of G. Using the definition (9), we obtain the equalities

Since the group F is commutative, we have the relations

as follows from (8). This proves that the classes of $c_{\sigma'}$ and $c_{\sigma}$ in $H^2(G,F)$ coincide.

Finally, let $\mathscr{E}= (\Gamma , \iota , \pi )$ and $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ be isomorphic $\tau$-extensions, let $u:\mathscr{E}\rightarrow \mathscr{E}'$ be a morphism of $\tau$-extensions, and let $\sigma$ be a section of the mapping $\pi$. The mapping $u\circ \sigma$ is a section of the mapping $\pi '$, and we have $c_{\sigma}=c_{u\circ\sigma}$. The class of $c_{\sigma}$ in $H^2(G,F)$ therefore depends only on the class of $\mathscr{E}$ in Ex$_{\tau}(G,F)$.

We denote by $\Theta_{\tau}$ : Ex$_{\tau}(G,F)\rightarrow H^2(G,F)$, or more simply Θ, the mapping that sends the class of a $\tau$-extension $(\Gamma , \iota , \pi )$ to the class of the 2-cocycle $c_{\sigma}$, where $\sigma$ is a section of the surjection $\pi$.