What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?

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.

• I don't have an answer, but just a comment regarding the analogous q'n for quasicoherent sheaves: if $\mathscr{X}$ is a tame DM-stack, then (by definition) the derived functors $\mathrm{R}^i \pi_\ast \mathscr{F}$ vanish for every quasicoherent sheaf $\mathscr{F}$ on $\mathscr{X}$ and every $i>0$, and any DM-stack over a field of characteristic $0$ is tame. – skd Sep 27 '18 at 22:34
• Even in the case where $Y$ is a point, so $\mathcal X$ is $BG$, you will be in trouble if $\ell$ divides $|G|$, as the cohomology of $BG$ is group cohomology which won’t match. But this should be your only obstruction by proper base change under whatever hypotheses are needed to ensure $\mathcal X_x$ is a single point. – Will Sawin Sep 28 '18 at 0:45
• Ah great! This is what I was hoping for - does anyone know what conditions on $\mathscr X$ make $\mathscr X_x = B \mathrm{Stab}_x$? Also, does proper base change work when $\ell \mid |G|$ (but isn't equal to the field characteristic)? I'd love to read a proof of these sorts of basic etale cohomology theorems which work for stacks. – dorebell Sep 28 '18 at 2:04

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.