Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X\_L$ for a finite extension $L$ of $K$. This gives a cocycle condition on a quasi-coherent sheaf on $X\_L$ to descend to $X$ (see for example page 139, section 6.2 of "Neron Models" by S. Bosch and others).


I am looking for an analogous result for the absolute Galois group i.e., if we take a coherent sheaf $E$ on $X\_{\bar{K}}$ which satisfies the analogous cocycle condition for any pair of elements $\sigma, \tau \in \mathrm{Gal}(\bar{K}/K)$, does there exist a (quasi)-coherent sheaf $F$ on $X$ such that $F \otimes\_K \bar{K} \cong E$?


Any hint/reference on this topic will be most helpful.



Galois descent for absolute Galois group
You need a continuity condition on the cocycles, otherwise it is probably false. The coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$. Now apply finite descent.


Added: The continuity condition just says that the descent datum splits over a finitely generated field extension of the base field. There is an elementary discussion of such things [here](http://www.jmilne.org/math/CourseNotes/AG16.pdf)


