I'm wondering what the "right" notion of equivariant cohomology is for something like étale cohomology or coherent cohomology, stuff which is expressible as derived functors of global sections of sheaves, in particular when $G$ is discrete (I think generally the nondiscrete case is more subtle, as then this doesn't even work for usual topological cohomology).

Can one simply take derived functors of invariant global sections of $G$-sheaves? Is there somewhere something like this is written down, with the usual desirable formal properties worked out? (functoriality, various long exact sequences, Hochschild-Serre spectral sequence, etc.) It seems to me like by Tohoku you should be able to get all of this in some generality, but I don't want to reinvent the wheel on something which seems like it has probably been considered. I'm relatively confident everything should work out if $G$ is finite, but a little worried about $G$ infinite.

I'm also curious about the same for algebraic de Rham cohomology; can one take hypercohomology of the de Rham complex with a $G$-action and get what one would expect?

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