Hello,

Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I understand correctly (I don't understand much), I have a functor $Sh(X_{et}) \to Sh(\overline{X}_{et})^{\Gamma}$, from etale sheaves on $X$ to etale sheaves on $\overline{X}$, equivariant w.r.t. the action of $\Gamma$ on $\overline{X}$.

My question is whether this is an equivalence of categories, and whether you can give a reference for this.

Thank you, Sasha