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Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I understand correctly (I don't understand much), I have a functor $Sh(X_{et}) \to Sh(\overline{X}_{et})^{\Gamma}$, from etale sheaves on $X$ to etale sheaves on $\overline{X}$, equivariant w.r.t. the action of $\Gamma$ on $\overline{X}$.

My question is whether this is an equivalence of categories, and whether you can give a reference for this.

Thank you, Sasha

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Hey Sasha! I don't think this has to be an equivalence of categories. If $X$ is a smooth projective variety, then the Hochschild-Serre spectral sequence gives us the sequence: $0\rightarrow Pic(X)\rightarrow Pic(\bar{X})^\Gamma\rightarrow Br(k)$, where it is known that the last map is not always zero. Using that $Pic$ describes invertible sheaves, it seems that sometimes the category $Sh(\bar{X}_{et})^\Gamma$ is bigger. A similar situation occurs when you go up one notch and consider $O_X$-algebras. There you can have equivariant Azumaya algebras that are not base-change. – Dror Speiser Feb 9 '12 at 16:34
@Dror: I don't know details, but my general feeling says that your argument does not contradict my statement? In my understanding, Serre-Hohcschild follows from interpretation of the unstructured global sections $Sh_G (Y) \to Ab$ as composition of $\Gamma: Sh_G (Y) \to G-mod$ with invariants $inv: G-mod \to Ab$; Where usually we already understand that $Sh_G (Y)$ is equivalent to $Sh(G\Y)$, for the $G\Y$ we have in mind. I might be wrong. – Sasha Feb 9 '12 at 17:15
up vote 2 down vote accepted

See SGA 7, XIII, Rappel 1.1.3; see also Geisser, Weil-étale cohomology over finite fields, Lemma 2.1 b).

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Thank you for the references! – Sasha Feb 10 '12 at 9:13

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