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let $G$ be an abelian group (can also assume $G=\mathbb{Z}^n$ for some positive integer $n$). Let $X\stackrel{g}{\rightarrow} Y$ be a $G$-covering where $X,Y$ are schemes, or topological spaces or sites. Is there any spectral sequence to compute invariants of cohomology, $H^i(Y,g_{*}F)^G$ where $F$ is an abelian sheaf? I would like to say that invariant line bundles, corresponding to $H^1(X,\mathbb{G}_m)^G$, are actually elements of $ H^1(Y,\mathbb{G}_m)$.

Thanks

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I must be misunderstanding something, because the only $G$-action I can see on $H^i (Y, g_* F)$ is trivial... –  Zhen Lin Jan 9 '13 at 14:23
    
@Zhen Lin why? the sheaf $g_{*}F$ is endowed with a $G$ action, for example take the projection $f:Y\times G \rightarrow Y$ and $F=\mathcal{O}_{Y\times G}$ –  kekki Jan 9 '13 at 15:24

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