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Let $X$ be a smooth, projective curve defined over a finite field $k$. We denote by $\overline{X}$ it's extension to the algebraic closure $\overline{k}$. All the questions are about coherent sheaves.

Under what conditions can we say that a sheaf $\mathcal{F}$ on $\overline{X}$ comes, by tensorization, from a sheaf on $X$?

Is it true that if $\mathcal{F}$ on $\overline{X}$ is stable under the Frobenius morphism then it comes from a sheaf on $X$?

What about the extensions of two sheaves on $X$? Namely, let $\mathcal{F,G}$ be two sheaves on $X$. Is it true that the (sheaf)extensions of these two sheaves on $X$ are the extensions of $\mathcal{F}\otimes\overline{k}$ and $\mathcal{G}\otimes\overline{k}$ that are stable under the Frobenius?

What if, for two sheaves $\mathcal{F},\mathcal{G}$ on $X$ we have an extension $\mathcal{E}\otimes\overline{k}$ of $\mathcal{F}\otimes\overline{k}$ and $\mathcal{G}\otimes\overline{k}$, is it true that $\mathcal{E}$ is an extension of $\mathcal{F},\mathcal{G}$?

The two last questions look to me like some sort of descent for morphisms at the derived categories level.

Edit: I forgot to specify that "sheaf"="coherent sheaf" in this topic.

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By 'stable', do you mean that the Frobenius acts as the identity on $\mathcal{F}$? Surely the only such sheaf on $\overline{X}$ with this property is the sheaf which is constantly equal to $\mathbb{F}_p$. –  Matthew Morrow Jun 2 '10 at 20:27
    
(ah, or direct sums of copies of this constant sheaf! Or maybe there are more interesting examples which I am missing.) –  Matthew Morrow Jun 2 '10 at 20:29
    
I was thinking more like the Frobenius applied to the sheaf is isomorphic to the sheaf of depart. So for example, if you tensorise a sheaf with $\overline{k}$ then the Frobenius acts on the second component and it is an isomorphism so you get the invariance... am I saying something stupid? Maybe this is not called Frobenius... I'm thinking more in terms of equations and relations and Frobenius acting on the coefficients (you can express any sheaf as a finitely generated graded module). –  Dragos Fratila Jun 2 '10 at 23:22
    
Could you give an example of what you mean by a (nontrivial) sheaf that is stable under the Frobenius (explicit equations and an action on a graded module are fine with me). –  Karl Schwede Jun 3 '10 at 7:46
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Need "continuity" condition on Frobenius descent datum. For coherent sheaves, amounts to finite-order condition after descent to finite intermediate field. For etale sheaves (of sets, abelian groups, etc.), amounts to distinction between "Weil sheaves" (in the sense of Deligne's Weil II) and usual etale sheaves. The answer to your 2nd question on extensions is surely "no", and for the 1st question on extensions you should formulate it in terms of a comparison of Ext^1, sheaf-Ext^1, and cohomologies thereof (depending on what flavor of sheaves you have in mind, which you don't make precise). –  BCnrd Jun 4 '10 at 10:58
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