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Are there any reasonable hypotheses on a map $f: X \to Y$ and a sheaf $E$ on $Y$ so that $f_* f^* E \cong E$?

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    This is basically a duplicate http://math.stackexchange.com/questions/6406/pullback-and-pushforward-isomorphism-of-sheaves (except mine is pushforward then pullback)2011-07-06
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    @trony, that's not true. That requires the hypothesis $f_\ast \mathcal{O} = \mathcal{O}$.2011-10-22

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