TITLE: When can cohomology be calculated on the coarse moduli space?
QUESTION [12 upvotes]: 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?

REPLY [5 votes]: (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.