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I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?

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I sort of answered this:… –  Ryan Reich Jun 11 '12 at 2:25
I don't understand. You can make an étale sheaf out of any group. –  S. Carnahan Jun 11 '12 at 10:03

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