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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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    $\begingroup$ For the étale topology, you may want to look into SGA 5, Exposé X by Grothendieck; or, something more condensed, like Lei Fu, Étale cohomology theory, 9.1. $\endgroup$
    – A.B.
    Commented Jun 15, 2021 at 23:15
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    $\begingroup$ You could take the cohomology of the quotient stack $[X/G]$, or more or less equivalently the cohomology of the simplicial scheme $X\times EG_\bullet/G$. That seems like it would be the "right" thing. $\endgroup$
    – Donu Arapura
    Commented Jun 15, 2021 at 23:56
  • $\begingroup$ I guess the stacky version should be the same as what I'm saying because the category of (adjective) sheaves on it should be the same as the equivariant sheaves on $X$ basically tautologically. don't know much about stacks, will the space $[X/G]$ be well-behaved even when $G$ is infinite discrete? i.e. enough to get the formalism I'd like in terms of Hochschild-Serre, etc. $\endgroup$
    – xir
    Commented Jun 16, 2021 at 0:56
  • $\begingroup$ @A.B. thanks those are good references for that case! $\endgroup$
    – xir
    Commented Jun 16, 2021 at 0:57
  • $\begingroup$ @xir Actions by infinite discrete groups do not commonly appear in algebraic geometry. What is the example you are trying to understand? $\endgroup$
    – Dan Petersen
    Commented Jun 16, 2021 at 6:11

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I am very far from knowledgeable about this, but having once asked a similar question elsewhere I was told that taking the derived category of equivariant sheaves is not the correct approach but that I should look at this book:

Bernstein, Joseph; Lunts, Valery.
Equivariant sheaves and functors.
Lecture Notes in Mathematics, 1578.

It may be worth a look based on your question.

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  • $\begingroup$ the first page of the introduction of this book and its section 8 specifically state that $D(Sh_G)$ is the correct thing to consider when $G$ is discrete, and makes reference to ch. 5 in tohoku as I expected, which discusses the topological case, and ch. 4 discusses modules over rings (i.e. affine coherent case). $\endgroup$
    – xir
    Commented Jun 16, 2021 at 13:32
  • $\begingroup$ likely tohoku can be adapted to the settings i'm interested in (in particular i think infty-categorical language makes the passage from rings to schemes formal), but I was hoping for a more precise reference in modern language. $\endgroup$
    – xir
    Commented Jun 16, 2021 at 13:32

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