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user127776
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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? ($i>1$$1<i<dim(X)$)

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? ($i>1$)

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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user127776
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Killing cohomology of structure sheaf by pullback along Frobenius and finite etale covers

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? ($i>1$)