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On a smooth projective variety $X$ over a finite field, you can pullback any element in $H^1(X,\mathcal{O}_X)$ by a combination of Frobenius and finite etale cover so it gets killed. In order to prove it I consider an element $\alpha$ in the cohomology group. This corresponds to an extension of $\mathcal{O}_X$ by itself. Since this cohomology group is finite and Frobenius acts on it, there is some $n$ and $m$ ($m>n$) such that $F^{*n}\alpha=F^{*m}\alpha$. By Prop 2.3 this means that you can pullback the extension $F^{*n}\alpha$ so it becomes a trivial vector bundle. Now the short exact sequence corresponding to $F^{*n}\alpha$ consists of three trivial vector bundles so it splits. This means pullback of $F^{*n}\alpha$ to some finite etale cover is trivial.

My question: is it possible to kill elements in $H^i(X,\mathcal{O}_X)$ by a combination of pullback along Frobenius and finite etale covers? ($1<i<dim(X)$)

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    $\begingroup$ yes you can. look at the paper of Hochster and Huneke for a generalisation: projecteuclid.org/download/pdf_1/euclid.bams/1183556250 $\endgroup$
    – ali
    Commented Jan 23, 2021 at 6:47
  • $\begingroup$ This seems like an expository paper. I'm not sure whether by finite and surjective they mean what I mean. I specifically require it to be a composition of Frobenius and finite etale covers. $\endgroup$
    – user127776
    Commented Jan 23, 2021 at 10:02
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    $\begingroup$ Take an ordinary smooth projective Calabi-Yau hypersurface $X \subset \mathbf{P}^{n+1}$ of big dimension (probably $n\geq 2$ is enough). Then $X$ will be simply connected as $n$ is big, so finite etale covers are useless. Ordinarity means the Frobenius acts bijectively on the $1$-dimensional vector space $H^n(X,\mathcal{O}_X)$, so that doesn't help either. $\endgroup$
    – Anonymous
    Commented Jan 23, 2021 at 17:30
  • $\begingroup$ I see so $i=dim(X)$ is problematic. $\endgroup$
    – user127776
    Commented Jan 24, 2021 at 3:03
  • $\begingroup$ This has little to do with whether $i$ equals $\dim X$, as you can take a product of any of these with $\mathbf P^N$. I think that already gives counterexamples for each $i \geq 2$ in each dimension $d \geq i$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 24, 2021 at 3:36

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