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Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that

$H^i(\mathcal{X},F) = H^i(X,\pi_\ast F)$.

Is there a simple example where this fails? Are there easy conditions where this is true?

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up vote 10 down vote accepted

Let k be a field and your DM-smack be [Spec(k)//G] for a trivial action of a group G. A sheaf on this stack is roughly a sheaf with a group action, and cohomology is group cohomology. If you consider a sheaf where multiplication by |G| is invertible, then group cohomology vanishes in high degrees, but otherwise there is a lot of information not coming from the cohomology of Spec(k).

EDIT: Under amenable circumstances when the left adjoint $\pi^\*$ is exact, you can get a sufficient condition for your isomorphism to hold. (I'm not sure what kind of topology you want to work with, or whether you're working with simply sheaves of abelian groups or sheaves of $\mathcal{O}$-modules or quasicoherent modules or...) In this case, $\pi\_\*$ preserves injectives and so there's a Grothendieck spectral sequence $$ H^p(X, {\mathbb R}^q \pi\_* F) \Rightarrow H^{p+q}(\mathcal{X},F) $$ and so a sufficient condition is for the higher direct image functors $\mathbb{R}^q \pi\_* F$ to vanish for q > 0. These are the sheaves associated to the presheaves $U \mapsto H^q(\pi^{-1} U, F)$, and often their stalks are related to the cohomology of the fibers of $\pi$ - which are, in this case of a DM-stack, pretty much the group cohomology of the stabilizers.

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(This mostly just adds some references to Tyler's answer)

$\newcommand{\X}{\mathcal X}$ As Tyler said, if $F$ is a quasi-coherent $\cal O$-module, you get the Grothendieck spectral sequence

$$H^p(X,R^q\pi_*(F))\Rightarrow H^{p+q}(\X,F)$$

so it suffices to impose the condition that $R^q\pi_*=0$ for $q>0$ (i.e. that $\pi_*$ is exact). This is exactly the condition that $\X$ is tame. See Abramovich-Olsson-Vistoli's Tame stacks in positive characteristic. In particular, DM stacks in characteristic 0 are always tame.

Note that it didn't matter that $\pi$ was a coarse space map. We just needed that $\pi_*$ is exact (that "$\pi$ is cohomologically affine"). In particular, the isomorphism holds when $F$ is quasi-coherent and $\pi:\mathcal X\to X$ is a good moduli space.

This is asserted in the first paragraph of the paper, but I don't see how to prove it. Perhaps somebody could clarify in a comment.

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Thanks a lot for the references, although I am (fortunately) not confused about the original question anymore. Your question about tame DM-stacks in char 0 seems to be answered by Theorem 3.2. in A-O-V, i.e. tame <=> all aut. grp. schemes at geometric points are linearly reductive. When X is DM, these group schemes are just finite groups, and they are then linarly reductive if and only if the order of the group is not divisible by the residue characteristic. – Dan Petersen Jan 14 '11 at 9:40

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