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Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.

Question: Refering to Donu Arapura's answer here, how to see that the higher direct image sheaves $R^if_* F $ are locally constant? ( it was stated there for an Abelian constant sheaf, but is the Abelian group sheaf structure really crucial?

Intuitively that reminds me on Ehresmann's classical theorem that such smooth surj proper $f$ must be already a bundle, so it is expected that it should behave very "nice" with direct images.

But I nowhere in literature on EC found a proof of this statement on locally constance of $R^if_* F $. Is it immediate after certain sophisticated observation (which I missed to make up to now)?

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    $\begingroup$ Abelian group structure is needed just to define the higher direct images. And doesn't the answer you linked include a reference to the result? $\endgroup$
    – Will Sawin
    Commented Apr 28 at 14:10
  • $\begingroup$ @WillSawin: Right, beeing sheaf of Abelian groups is neccessary to have enough injectives to resolve so far I understand. Yes sorry, I overlooked, in Kiehl & Freitag's EC it's Thm 8.9, and another source rather nice elaborating this is Lei Fu's ECT, Cor 7.8.3. $\endgroup$
    – user267839
    Commented Apr 28 at 21:48

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