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Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$) equivalent to the vanishing of all $H^i(f^{-1}(s), L|_{f^{-1}(s)})$ for all $i \geq 0$ and all points $s \in Y$?

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    $\begingroup$ Do you want the morphism to be proper? That already fails for open immersions. $\endgroup$
    – Jason Starr
    Commented Aug 16 at 4:30
  • $\begingroup$ @JasonStarr Yes, the morphism is proper, sorry, I fixed this in the question $\endgroup$
    – Yellow Pig
    Commented Aug 16 at 7:59

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