Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$). Then, does there exist a finite extension $L$ of $k$ and a coherent sheaf $E_L$ on $X_L$ such that the pull-back of $E_L$ to $X_{\overline{k}}$ is isomorphic to $E_{\overline{k}}$?
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4$\begingroup$ That is true, and you can prove it in a very explicit fashion by taking a locally free presentation of $E_{\overline{k}}$ by locally free sheaves of the form $\bigoplus \mathcal{O}(-d_i)$, resp. $\bigoplus \mathcal{O}(-e_j)$. The "matrix" of this presentation is a finite sequence of maps $\mathcal{O}(-e_j)\to \mathcal{O}(-d_i)$. Define $L$ to be the subextension of $\overline{k}/k$ generated by all coefficients of those maps with respect to the standard monomial basis. A more thorough answer follows from EGA $IV_3$, Sections 8, 11.2, and 11.3. $\endgroup$– Jason StarrCommented Aug 23, 2017 at 10:54
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2$\begingroup$ Arguments of this type are often called "limit arguments". They come up in the literature when proving some scheme (or algebraic space, or stack) is locally finitely presented. So if you scan Artin's papers on algebraic spaces and stacks, or Olsson's papers, Hall-Rydh, Lieblich, etc., you will find plenty of examples of this technique in action. $\endgroup$– Jason StarrCommented Aug 23, 2017 at 10:58
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$\begingroup$ @JasonStarr Thank you for the answer. $\endgroup$– RonCommented Aug 23, 2017 at 11:44
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2$\begingroup$ For the last sentence of Jason Starr's first comment, the most relevant much wider context on "descent through direct limits" for such sheaf data (hypotheses on $X$ and $k$ can be weakened a tremendous amount) is EGA IV$_3$ 8.5.2(ii) (with the setup in 8.2.2 applying to the collection of $X_L$'s as the $S_{\lambda}$'s for $S_0 = X$) as well as 8.5.2.5 and 8.5.5, although as Starr notes the rest of section 8 as well as 11.2 and 11.3 supply many more very useful results on this theme. $\endgroup$– nfdc23Commented Aug 23, 2017 at 12:45
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