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If $X$ is a scheme (maybe with conditions), I'm pretty sure that the ($\ell$-adic/de Rham) rational cohomology $H^*(X,\mathcal{F}$) of an $\ell$-adic sheaf/holonomic $D$-module $\mathcal{F}$ vanishes above some degree $d(X)<\infty$. Say that $X$ has finite cohomological dimension. This does not hold for Artin stacks, e.g. $H^*(BG,k)$ lives in arbitrarily high degrees for most positive-dimensional $G$. Thus I imagine there is a theorem of the form

Theorem(?): For all Deligne-Mumford stacks (or maybe only algebraic space?) $X$ (with conditions?), $X$ has finite cohomological dimension.

Question: Does such a theorem exist, either in the $\ell$-adic or the $D$-module form? If so, what is a good reference?

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  • $\begingroup$ What if $X$ is the disjoint union of complex projective spaces of each dimension? $\endgroup$
    – Faris
    Commented Mar 23, 2021 at 10:47
  • $\begingroup$ @Faris Yeah, I guess $X$ should should be of finite type $\endgroup$
    – user42024
    Commented Mar 23, 2021 at 10:50
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    $\begingroup$ And it is also necessary to make assumptions on the cohomological dimension of the base field! Let's maybe assume the base field is algebraically closed. Then I think the statement is true for all DM-stacks $X$ of finite type. This is definitely true for quotient stacks by etale group schemes: you get a spectral sequence converging from the cohomology of a (finite) group with coefficients in the cohomology of a finite type scheme.. I guess it should also be true that if $U\rightarrow X$ is an etale cover, then rat. cohomological dimension of $X$ is bounded by the cohomological dimension of $U$ $\endgroup$
    – user42024
    Commented Mar 23, 2021 at 10:56
  • $\begingroup$ but I can't recall the argument $\endgroup$
    – user42024
    Commented Mar 23, 2021 at 10:56
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    $\begingroup$ I explained this to a graduate student a couple of weeks ago. There is a quasi-finite, proper $1$-morphism from the stack to a "coarse moduli space". Etale locally, this morphism is the coarse moduli space of a finite quotient stack (you can find a proof in Abramovich-Vistoli). Thus, by the argument in the previous comment, the higher direct images vanish for "rational" $\mathbb{Q}_\ell$-coefficients. Combined with the Leray spectral sequence, cohomological dimension of the stack equals cohomological dimension of the coarse moduli space. $\endgroup$
    – Jason Starr
    Commented Mar 23, 2021 at 16:23

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