Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space. 


What are some general results on the relationship between $H^i\_{\mathrm{et}}(\mathscr{X}, \underline{\mathbf{Z}\_\ell})$ and $H^i\_{\mathrm{et}}(X, \underline{\mathbf{Z}\_\ell})$? By the usual spectral sequence argument, I guess I'm asking for what some general results are about the pushforwards $R^i \pi\_\* \underline{\mathbf{Z}\_\ell}$. If I'm not mistaken, proper base change should tell us that these are (constructible? lcc?) $\ell$-adic sheaves with stalks $H^i\_{\mathrm{et}}(\mathscr{X}\_x, \underline{\mathbf{Z}\_\ell})$. 


I'm happy for results that work in significantly less generality, or to know what some interesting conditions on $\mathscr{X}$ are which make this question easier. Conversely, I'd love to hear something that works when $\underline{\mathbf{Z}\_\ell}$ is replaced with some other (lcc etc.) $\ell$-adic sheaf. 


My motivation for this question came from the case where $\mathscr{X} = [Y/G]$ for $Y$ a smooth projective variety (even a hypersurface) over $\mathbf{C}$ and $G$ a finite cyclic group acting with non-discrete fixed points, and I wanted to compute torsion in the singular cohomology with $\mathbf{Z}$-coefficients. 



What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?
The result you want is this.



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> > Let $f : \mathscr{X} \to S$ be a proper tame DM stack with $S$ a scheme, and $g : S' \to S$ any morphism of schemes. Let $\mathscr{F}$ be a torsion sheaf on $\mathscr{X}$. Then the natural base change morphism
> >  $$g^\ast R^i f\_\ast \mathscr{F} \to R^if'\_\ast g'^\ast \mathscr{F}$$
> >  is an isomorphism.
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For a proof, see Theorem A.0.8 of [Abramovich-Corti-Vistoli](https://arxiv.org/pdf/math/0106211.pdf). They also provide a description of the stalks of the higher pushforwards, which is something we discussed yesterday.


