# Deligne finitude and finiteness of etale cohomology

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.

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.
• That question assumes $X$ is proper and Deligne theorem is about $Rf_!$ not $Rf_*$. Commented Sep 16, 2023 at 19:17
• The $X$ of that question does not refer to your $X$ in this context, but to Spec $k$, which is proper over itself. Commented Sep 16, 2023 at 19:19
• See Deligne’s theorem here which refers to $R f_*$, not $R f_!$: matematicas.unex.es/~navarro/res/sga/… Commented Sep 16, 2023 at 19:21