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.

  • $\begingroup$ Isn't this just a special case of faithfully flat descent? $\endgroup$ – Denis Nardin Jul 20 '17 at 20:07
  • $\begingroup$ @DenisNardin I think so, but not completely sure. So, I was trying to find a reference. $\endgroup$ – user45397 Jul 20 '17 at 20:19
  • $\begingroup$ @DenisNardin: No. The tensor square of an infinite-degree algebraic extension is a beast which is not well-controlled by Galois data alone. One must impose a "continuity" condition, as noted in anon's answer below. $\endgroup$ – nfdc23 Jul 21 '17 at 0:31
  • $\begingroup$ @nfdc23 I thought that $\mathrm{Spec}(\bar{K}\otimes_K\bar{K})\cong \bar{K}\times G$ when $G$ is seen as a group scheme (the spectrum of the Hopf algebra of locally constant functions from $G$ to $\mathbb{Z}$). Am I wrong? $\endgroup$ – Denis Nardin Jul 21 '17 at 7:49
  • $\begingroup$ @DenisNardin Presumably the continuity condition shows up as the condition that the map be defined by algebraic functions on that group scheme, which are necessarily locally constant, and not arbitrary functions on the group. $\endgroup$ – Will Sawin Jul 21 '17 at 11:44

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

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    $\begingroup$ There are essentially always discontinuous 1-cocycles valued in a discrete Galois module (e.g., valued in $\mathbf{Z}/(2)$ for a field with infinitely many non-isomorphic quadratic Galois extensions), so that provides non-effective descent data if one doesn't impose the continuity condition (since the continuity condition is a consequence of effective descent). $\endgroup$ – nfdc23 Jul 21 '17 at 4:14
  • $\begingroup$ @anon As I am beginner in the topic, I am not very familiar with the terminology. Could you please elaborate a bit on what you mean by continuity condition or simply give a reference. $\endgroup$ – user45397 Jul 21 '17 at 8:26
  • $\begingroup$ @user45397 The continuity should simply be that the action of $\operatorname{Gal}(\overline{K}/K) $ by pullback on the set $S$ pairs of an open set $U$ and a section of $E(U)$, viewed as map $\operatorname{Gal} \times S \to S$, is continuous using the standard (profinite) topology on $\operatorname{Gal}(\overline{K}/K)$ and the discrete topology on $S$. $\endgroup$ – Will Sawin Jul 21 '17 at 11:39

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