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If $f: X \to Y$ has geometrically connected fibres, $f_*\mathbb{G}_m = \mathbb{G}_m$ as Zariski sheaves. Does it also hold as ├ętale sheaves?

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Your claim is false even for Zariski topology. Please clarify the hypothesis. –  user18119 Jan 29 '13 at 15:28

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