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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.

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    $\begingroup$ 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. $\endgroup$ – skd Sep 27 '18 at 22:34
  • $\begingroup$ 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. $\endgroup$ – Will Sawin Sep 28 '18 at 0:45
  • $\begingroup$ 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. $\endgroup$ – dorebell Sep 28 '18 at 2:04
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The result you want is this.

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.

For a proof, see Theorem A.0.8 of Abramovich-Corti-Vistoli. They also provide a description of the stalks of the higher pushforwards, which is something we discussed yesterday.

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  • $\begingroup$ @dorebell I prefer not to accumulate reputation for answers that give direct references. $\endgroup$ – Ben Lim Sep 28 '18 at 23:06
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    $\begingroup$ A more comprehensive reference is Corollary 5.6.7 of the following: Kai Behrend, Derived l-adic categories for algebraic stacks. Memoirs of the American Mathematical Society Vol. 163, 2003. This is available from Behrend's webpage: math.ubc.ca/~behrend/ladic.pdf $\endgroup$ – Jason Starr Sep 30 '18 at 0:57

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