This probably is a very straightforward question. Does Deligne finitude imply etale cohomology with $\mu_l^{\otimes n}$ ($l$ is invertible) for finite type schemes over a finite field is finite? This is Lemma 3.12 here. I can see how it can imply cohomology with compact support is finite because the theorem involves proper pushforward but not sure about the regular cohomology.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
See the answers here: https://mathoverflow.net/questions/76069/finiteness-of-étale-cohomology-groups . Because $k$ is finite, a constructible sheaf on (Spec $k)_{et}$ has finite cohomology groups. Deligne’s theorem says that the higher pushforwards of a constructible etale sheaf on $X$ via $f: X \to$ Spec $k$ are constructible, and thus have finite cohomology groups. Appealing to the Leray spectral sequence, which states that $H^i($(Spec $k)_{et}, R^j f_* \mu_l^{\otimes n}) \to H^{i + j}(X_{et}, \mu_l^{\otimes n})$, we obtain finiteness of the latter.
-
$\begingroup$ That question assumes $X$ is proper and Deligne theorem is about $Rf_!$ not $Rf_*$. $\endgroup$ Commented Sep 16, 2023 at 19:17
-
1$\begingroup$ The $X$ of that question does not refer to your $X$ in this context, but to Spec $k$, which is proper over itself. $\endgroup$– Vik78Commented Sep 16, 2023 at 19:19
-
2$\begingroup$ See Deligne’s theorem here which refers to $R f_*$, not $R f_!$: matematicas.unex.es/~navarro/res/sga/… $\endgroup$– Vik78Commented Sep 16, 2023 at 19:21