Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu\_n$ (the group of $n$th roots of unity). 
The étale cohomology groups of X are therefore equipped with an action of $\mu\_n$.


Now, let's suppose that the action of $\mu\_n$ on $X$ extends to an action of $\mathbf G\_m$.
Then, the analytic space $X(\mathbf C)$ is equipped with an action of $\mu\_n$ which extends to an action og $\mathbf C^\times$. As $\mathbf C^\times$ is connected, one concludes that the action of $\mu\_n$ on the singular cohomology of $X(\mathbf C)$ is trivial (the multiplication by each element $\xi\in \mu\_n$ is homotopic to the identity).


Going backwards to étale cohomology, using a comparison theorem between étale and singular cohomology, one concludes that the action of $\mu\_n$ on the étale cohomology of $X$ is trivial.


Is there a direct proof that the action on the étale cohomology groups is trivial without having to refer to the topological case? 



Algebraic proof without using comparison theorem for étale cohomology
Smoothness of $X$ is not needed (neither for the comparison isomorphism nor for the result in question). Let $X$ be any quasi-separated scheme over a separably closed field $k$, equipped with an action by a connected $k$-group scheme $G$ of finite type. Let $n > 0$ be an integer not divisible by the characteristic of $k$ and choose an integer $i \ge 0$. Then we want to show that the action of $G(k)$ on ${\rm{H}}^i(X, \mathbf{Z}/(n))$ is trivial (using etale cohomology here). 


[The hypothesis on $n$ is necessary because if $n = p = {\rm{char}}(k)>0$ and $X = {\rm{Spec}}(A)$ is affine then the effect of $G(k)$ on ${\rm{H}}^1(X, \mathbf{Z}/(p)) = A/\wp(A)$ (with $\wp(f) = f^p-f$) is the induced action from the $G(k)$-action on $A = \Gamma(X,O\_X)$, and this is generally nontrivial (e.g., $X = G = \mathbf{A}^1\_k$ with the translation action corresponding to $c.f(t) = f(t+c)$ on global functions (for $c \in G(k)$).]


By a spectral sequence argument using a covering by quasi-compact $G$-stable open subsets we may reduce to the case when $X$ is quasi-compact (and quasi-separated).
It is harmless to make the radiciel extension from $k$ to its algebraic closure ("topological invariance" of etale cohomology), so we may assume that the separably closed $k$ is even algebraically closed. We may then replace $G$ with $G\_{\rm{red}}$ so that $G$ is smooth. 


Let $f:G \times X \rightarrow G$ be the projection map. The hypothesis on $n$ and smoothness of $G$ allow us to apply the smooth base change theorem to conclude that $\mathscr{F} = {\rm{R}}^if\_{\ast}(\mathbf{Z}/(n))$ is the constant sheaf on $G$ attached to ${\rm{H}}^i(X,\mathbf{Z}/(n))$.
Consider the action automorphism 
$$\alpha: G \times X \simeq G \times X$$
defined by $(g,x) \mapsto (g, gx)$. This commutes with $f$, and so induces an automorphism $[\alpha]$ of $\mathscr{F}$ on $G$. The effect on the stalk at $g \in G(k)$ is the $g$-action on ${\rm{H}}^i(X, \mathbf{Z}/(n))$ that we want to be trivial. But $\mathscr{F}$ is a *constant* sheaf on a *connected* scheme, so the effect on $\mathscr{F}$ of any automorphism is uniquely determined by the effect on a single stalk. Looking on the stalk at $g=1$ thereby shows that $[\alpha]$ is the identity automorphism. Now pass to the effect of $[\alpha]$ on the stalk at any $g \in G(k)$ to conclude.


