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user267839
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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)?

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 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)?

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)?

Source Link
user267839
user267839
  • 6k
  • 2
  • 11
  • 42

Higher direct images of locally constant etale sheaf under smooth proper map locally constant

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 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)?