# Galois action on the derived category of $l$-adic sheaves

Hi,

Suppose that $G$ is a group acting on a scheme $X$, and $F$ is an $l$-adic sheaf on $X$ EDIT: with an action of $G$ (thanks Torsten). Is it true that $R\Gamma(X,F)$ is well-defined as an object of the derived category of $G$-equivariant $l$-adic sheaves? I'm thinking about the case when $G$ = Galois group, but I'm also interested in the more general case.

Thanks!

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You have to have an action of $G$ also on $F$, i.e., a collection of maps $g^\ast F\to F$ fulfilling the obvious properties. In the case when $F$ is a constant sheaf there is automatically such an action but in general it is extra data. –  Torsten Ekedahl Jun 26 '11 at 6:23
By "$G$-equivariant $l$-adic sheaves" do you mean "$l$-adic $G$-modules"? Or you are working in the relative case $f:X\to Y$ and asking if $Rf_*F$ is a $G$-equiv. complex of sheaves on $Y$ (which has trivial $G$-action say)? –  shenghao Jun 26 '11 at 9:38
The answer is now yes, $\Gamma$ may be seen as a functori from $G$-sheaves to $G$-modules and one can define $R\Gamma(X,F)$ as its derived functor. It is easy to see that the forgetful functor from $G$-modules to modules takes this $R\Gamma(X,F)$ to the usual one (use Grothendieck's composite functor theorem). –  Torsten Ekedahl Jun 26 '11 at 15:43