Pullback and etale cohomology I am pretty sure I don't understand well the action of a pullback of some etale map on the first etale cohomology group. In fact, let $f : T \rightarrow X$ (etale map) be a $X$-torsor for some algebraic group $G$. It gives a cohomology class on $H^{1}_{et}(X, G)$. Now, I consider $g : X \rightarrow X$ be another etale map, and I look to $g^*(T) = T \times_X X$, which is a $X$-torsor (given by the second projection).
First example : when we take $X=T=\mathbb{C}-\{0\}$, $f : T \rightarrow X$ given by $z \mapsto z^n$, and $g : X \rightarrow X$ given by $z \mapsto z^m$, with $\gcd(n,m)=1$, some straighforward computations give that the map $g^* : H^{1}_{et}(X, G) \mapsto H^{1}_{et}(X, G)$ is the multiplication by $m$.
My problem : In general, we have that $g^*(T)= T \times_X X$ is a $X$-torsor for the second projection, with group action of $G$ given by the action of the first coordinate. Now, let $(U_i \rightarrow X)_i$ be a family which trivialize the $X$-torsor $T$. Then, denoting $U_{ij} := U_i \times_X U_j$, for $s_i \in U_i$, $s_j \in U_j$, there exists a unique $g_{ij} \in G(U_{ij})$ such that $s_j = g_{ij} s_i$, and this is how we identify the $X$-torsor $T$ to $[(g_{ij})_{ij}] \in H^{1}_{et}(X, G)$.
Now, let $U_i' := U_i \times_{X, g} X$. Let $s_i : U_i \rightarrow T$ a section ($s_i \in T(U_i)$). Then, if $(x_1, y_1) \in U_i \times_{X, g} X$, we have that $f(s_i(x_1))= (U_i \rightarrow X)(x_1) = g(y_1)$, and then we can construct a section $s'_i : U_i' \rightarrow g^*(T)$ by $(x_1, y_1) \mapsto (s_i(x_1), y_1)$. It gives us an application $\pi_i : T(U_i) \rightarrow g^*(T)(U'_i)$. As the action of $G$ on $g^*(T)$ is given by the action of $G$ on the first coordinate, the previous application is compatible with the action of $G$, where the map $\text{res}_i : G(U_i) \rightarrow G(U'_i)$ is given by the "restriction" map (the composition with the projection $U_i' = U_i \times_{X, g} X \rightarrow U_i$).
But then, if $s_j = s_i g_{ij}$, we have $\pi_j(s_j) = \pi_i(s_i) \text{res}_{ij}(g_{ij})$. Then, the class of $g^*(T)$ in $H^{1}_{et}(X, G)$ is $[\text{res}_{ij}(g_{ij})_{ij}]$ which is the same class as $[(g_{ij})_{ij}]$.
The "proof" is clearly false, otherwise $g^*$ would be trivial for every $g$... In fact, I never use $g$ in the proof.
Where is the mistake ? It's clearly wrong, but I don't see what is the point.
Thank you !
 A: Finally, I think the confusion come from the relation between the $X$-torsor $T$ and the class of cohomology which is associated to $T$. We have, on one hand, that $T$ correspond to $[g_{ij}] \in H^1((U_i \rightarrow X)_i, G)$, and on the other hand, the transition functions of the pullback are given by $g_{ij} \circ f_{U_i}$ where $f_{U_i}$ is the projection $U_i \times_X T \rightarrow U_i$, and then $g^*(T)$ correspond to the cohomology class $H^1((U_i \times_X X \rightarrow X)_i, G)$ where $U_i \times_X X \rightarrow X$ is the second projection (we do not compose this morphism by $f$ !).
My confusion came to the fact that I thought that the covering $(U_i \times_X X \rightarrow X)_i$ was a refinement of $(U_i \rightarrow X)_i$, and then as $H^1(X, G) := \underset{\longrightarrow_{U}}{\lim}H^1(U, G)$ where $U$ go all over the etale covering of $X$, I thought that $[g_{ij}]$ was the same as $[(g_{ij} \circ f_{U_i})_{ij}]$. But actually, this is wrong, cause the covering $(U_i \times_X X \rightarrow X)_i$ is not the composition of the $f$ and the second projection, but is only the second projection, and then not, in general, a refinement of $(U_i \rightarrow X)_i$.
The example I took in my question permits to do a computation to understand this. We have that $g^*(T) = \mathbb{C} - \{0\}$ where the first projection onto $T$ is given by $z \mapsto z^m$ and the second projection onto $X$ is given by $z \mapsto z^n$, which is the same as $f : T \rightarrow X$, the structural morphism of the $T$-torsor onto $X$. Then, if $(i_{U_i} : U_i \rightarrow X)_i$ is a trivialization of $T$, the same etale family $(i_{U_i} : U_i \rightarrow X)_i$ is a trivialization of $g^*(T)$ cause the structural morphism of $g^*(T)$ as a $X$-torsor (given by the second projection !) is the same as the structural morphism of $T$ as a $X$-torsor (given by $f$). So, we have the same transition functions for $T$ and $g^*(T)$, but as the action of $G$ over $T$ is given by $z \cdot \overline{k} := z \zeta_{n}^{k}$ and the action of $G$ over $g^*(T)$ is given by $z \cdot \overline{k} := z \zeta_{n,m}^{k}$ where $\zeta_{n,m}$ is the unique $m$-th unity root on the group of $n$-th unity roots (cause $gcd(m, n)=1$), then, $g^*(T)$ corresponds to $[m \cdot g_{ij}] \in H^1((U_i \rightarrow X)_i, G)$ while $T$ corresponds to $[g_{ij}] \in H^1((U_i \rightarrow X)_i, G)$, and then we find that the action of $g^*$ on $T$ is the multiplication by $m$, by the identification between torsors and first etale cohomology group.
