# Finiteness result for higher direct image of $\ell$-adic sheaves

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.

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

$$Y$$ 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.

• Whoops, I must have not been thinking when I wrote the question as it was. I've edited it now, hopefully it is less trivial (or perhaps even correct).
– Meow
Jun 18 at 23:42
• @Meow Isn't the disjoint union of $\mathbb A^m$ for all $m$ an Artin stack? You need some kind of finiteness assumption on $X$. Jun 19 at 1:19
• Right, I should have included a finite type assumption, which has been edited in now.
– Meow
Jun 19 at 9:07
• @Meow See edits. Jun 19 at 14:41
• Thanks for being accomodating of my shifting the goalposts!
– Meow
Jun 19 at 23:52