Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf\_\*\mathbf{Q}\_\ell=0$ for all $q\gg 0$?


Note: by taking atlases, I think it is sufficient to let $X,Y$ be schemes.




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**Edit**: Will Sawin pointed out that the question as stated was obviously false, I've edited it to remove that false statement.



Finiteness result for higher direct image of $\ell$-adic sheaves
$Y$ admits a smooth surjective morphism from a scheme $Z$. Because smooth morphisms are locally of finite type, $Z \to Y$ is locally of finite type, and you can choose an open cover that covers $Y$ and then pass to a finite subcover to make $Z$ of finite type.


Because this morphism is smooth, by smooth base change the pullback of $R^q f\_\* \mathbf Q\_\ell$ to $Z$ is the pushforward of $\mathbb Q\_\ell$ from $Z \times\_Y X$ to $Z$. Because this morphism is smooth, it suffices to prove a bound for this pushforward.


If $Z \to Y$ is a schematic morphism (this might be a little stronger than the fibers being schemes) then $Z \times\_Y X$ is a scheme, also of finite type. Boundedness then follows from classical results - mod $\ell$ cohomology is a limit over the cohomology of neighborhoods, and these are finite type of bounded dimension.


